For this question, I would like to first define that someone is a Jew if they meet the halakhic definition. This definition is just for the sake of moving the discussion toward an answer.
Is it possible for a non-Jew to join Messianic Judaism?
In the same wiki to which you linked, there is a section on Converts to Judaism, which states that all "mainstream forms of Judaism today are open to sincere converts..." Even in the halakhic perspective, there are ways to become a full Jew. I would add that there has always been a path to becoming a Jew (think about Rahab and Ruth). And, since there are no ethnic barriers to Christianity (see the book of Acts), it would seem to me that there are no ethnic barriers for anyone to become a Messianic Jew.
"Messianic Judaism" is not a single denomination or sect; the term covers a broad range of beliefs and practices. Various groups exist; often the disagreements concern the extent toward which the Mosaic law must be followed today. I'm going to answer based on the views of two groups: the Messianic Jewish Alliance of America (MJAA), and the Union of Messianic Jewish Congregations (UMJC). 40-60% of Messianic Judaism's congregations are Gentiles, so this is an important question.
I'm going to sacrifice accuracy a bit to be concise: The MJAA tries to balance between Christianity and Judaism; if forced to distance themselves from one they tend to align themselves with evangelical Christianity.
From the MJAA Statement of Faith:
We recognize that Jewish people (physical descendants of Abraham through Isaac and Jacob, whether through the mother's or the father's blood-line) who place their faith in Israel's Messiah, Yeshua, continue to be Jewish according to the Scriptures (Romans 2:28-29). Gentiles who place their faith in Yeshua, are "grafted into" the Jewish olive tree of faith (Romans 11:17-25) becoming spiritual sons and daughters of Abraham (Galatians 3:28-29). ... We acknowledge that the New Covenant body of believers is composed of both Jews and Gentiles who have received Yeshua the Messiah as the Promised Redeemer. The "middle wall of partition" has been broken down and now we worship the God of Israel together (I Corinthians 12:13; Ephesians 2:13-14).
Concerning the acceptance of non-Jews into their congregations ("Messianic Gentiles"), Kesher Journal reports (emphasis mine):
Congregations are faced with internal division from having a membership that includes Jews, who take part in the destiny and specific promises to the Jewish people, and Gentiles, who do not. The official stance is that Gentiles and Jews are spiritually equal but distinct, and that Jews should be proud of being Jews, and Gentiles proud of being Gentiles. Nevertheless, the Jewish identity is clearly valorized, causing many Gentiles to strive for greater Jewishness through Jewish observance and search for Jewish roots. Since conversion for Gentiles is deemed unbiblical within the MJAA, these are the main options for Gentiles seeking a more Jewish identity. What helps increase a sense of unity between Jews and Gentiles within these rituals is the stress on the importance of being a "spiritual" Jew more than a "physical" Jew...
Messianic Jews within the MJAA will accept a Gentile into their congregations, but not allow conversion to Judaism; you will be a "Messianic Gentile." As Ellen Kavanaugh puts it concerning Christians who consider converting to Messianic Judaism,
Conversion implies a change in belief -- from one belief system to a different belief system. And therein lies the problem: To convert to Messianic Judaism would be to 'convert' to a community in which one was already [a] member; to convert to traditional Judaism would be to deny one's faith in Yeshua (or at least, deny the sufficiency of Yeshua alone).
The UMJC strives to be an authentic Jewish group, and be recognized as such by non-Messianic Judaism. They strive for "authentic Judaism."
They define Messianic Judaism as such:
Jewish life is life in a concrete, historical community. Thus, Messianic Jewish groups must be fully part of the Jewish people, sharing its history and its covenantal responsibility as a people chosen by God. At the same time, faith in Yeshua also has a crucial communal dimension. This faith unites the Messianic Jewish community and the Christian Church, which is the assembly of the faithful from the nations who are joined to Israel through the Messiah. Together the Messianic Jewish community and the Christian Church constitute the ekklesia, the one Body of Messiah, a community of Jews and Gentiles who in their ongoing distinction and mutual blessing anticipate the shalom of the world to come.
For a Messianic Jewish group 1) to fulfill the covenantal responsibility incumbent upon all Jews, 2) to bear witness to Yeshua within the people of Israel, and 3) to serve as an authentic and effective representative of the Jewish people within the body of Messiah, it must place a priority on integration with the wider Jewish world, while sustaining a vital corporate relationship with the Christian Church.
In the Messianic Jewish way of life, we seek to fulfill Israel's covenantal responsibility embodied in the Torah within a New Covenant context. Messianic Jewish halakhah is rooted in Scripture (Tanakh and the New Covenant writings), which is of unique sanctity and authority. It also draws upon Jewish tradition, especially those practices and concepts that have won near-universal acceptance by devout Jews through the centuries. Furthermore, as is common within Judaism, Messianic Judaism recognizes that halakhah is and must be dynamic, involving the application of the Torah to a wide variety of changing situations and circumstances.
Messianic Judaism embraces the fullness of New Covenant realities available through Yeshua, and seeks to express them in forms drawn from Jewish experience and accessible to Jewish people.
To again quote Kesher Journal:
In a recent discussion at the 2002 Lausanne Consultation on Jewish Evangelism (LCJE) Conference on Messianic Jewish Identity, a UMJC affiliated rabbi argued that Messianic Jews who do not keep kosher are in sin against God, denying their calling and identity... One of the implications of this stance is that observance of the traditions becomes a measure of spirituality and closeness with God, creating a spiritual hierarchy between those who are most observant relative to those who are less so. By keeping the law, Union congregations also hope to build bridges with observant parts of the Jewish community.
The UMJC, in contrast to the MJAA, requires "Messianic Gentiles" to observe the Jewish laws, and generally to convert fully -- becoming Messianic Jews.
From the Kesher Journal article once more:
In the past, Messianic congregations have generally defined themselves as a place where Jews and Gentiles worship together, witnessing to the unity of Jews and Gentiles. Many in the UMJC are seeing this as a defective definition. Congregations may have Gentiles, but they are not part of the definition of the congregation, which is to be a Jewish space. The congregation is not an adequate witness to the unity of Jews and Gentiles because the Gentiles there are called to live as Jews- the unity would only be demonstrated if Gentiles were to live as Gentiles.... The simple reality is that many Messianic Jewish leaders realize the kinds of congregations being built are unable to adequately express the Jewish life. They are not seen as authentically Jewish.
As with other issues, the UMJC's distinguishing mark is its concern for Jewish authenticity. One way to be more authentically Jewish is by adopting a conversion process, similar to other forms of Judaism.
In short, whether a Gentile can convert to Messianic Judaism depends in large part on the beliefs of the specific congregation to which he or she would like to join. In some congregations, you may be able to convert and become a Messianic Jew, while in others you will be accepted to the community, but as a "Messianic Gentile."
Although it can be done (from a halakhic perspective), it would be a bad idea to do so from a New Testament perspective. Paul advised against it.
18 Is any man called being circumcised? let him not become uncircumcised. Is any called in uncircumcision? let him not be circumcised. 19 Circumcision is nothing, and uncircumcision is nothing, but the keeping of the commandments of God. 20 Let every man abide in the same calling wherein he was called.
1 Cor 7
Anyone can join.
Your question alludes, that your understanding is one must be a Jew to join. Your question also alludes, that the movement has something to do with Judaism.
I have personal experience with converts and I can fill in some gaps based on the formation of your question.
If you want facts related to Judaism, I would council the thousands of year old faith itself, that is an Orthodox Jewish Rabbi to clear up any confusion you might have, in it's relation to Judaism.
My knowledge is the short history of the messianic group is not a sect of Judaism, as much as Protestant sect mixed with Jewish culture, not Judaic worship. Yes, there are some Jews in these groups, but they have converted to Evangelical Protestantism.
A solid example: One of the most popular groups, "Jews for Jesus" an associated group to this movement, founder was not a Jew, but an ordained Baptist preacher at the time of the founding, named Martin Rosen.
There were a number of converted Jews / cofounders who traveled with Martin to California during the "Jesus movement" of the 1960's.
One cofounder was James Bernstein. The actual man who had a jacket embroidered with the words "Jews for Jesus", for which Martin told him , thanks to his jacket, that would be the name of their outreach. (Not exactly extended to be a movement at the time)
Most Jewish cofounders like James (except for one) left the group and became Orthodox Christian Presbyters, leaving the vast majority of the group mostly Protestant and what slowly became a Protestant movement with many forms of worship, teaching,, preaching etc.
His book and story of his history with the Messianic movement is in the following link. Surprised by Christ: My Journey From Judaism to Orthodox Christianity by A. James Bernstein Link: https://amzn.com/1888212950
Your question also alludes, if not states that you are looking for a more Judaic form of Christian worship.
If you are looking for the most Judaic form of Christian worship with roots and history. History that has existed for nearly 2,000 years, look to Ethiopia. Ethiopian Orthodox Christianity is the most ancient and Judaic form of Christian worship in existence. Ethiopia was predominantly a Jewish state before converting to Christianity with the Queen Sheba having a strong relationship with King Solomon. Of course one of the wise men was Ethiopian, Philip converts the Ethiopian Eunuch in Acts. Ethiopia has the original Ark of the Covenant in their possession, and you will not convince them otherwise. All of the nearly 2,000 years of history is in place for one to research. There are a few Ethiopian Orthodox Churches in America as well. PS: Although Ethiopian worship is truly the "most" Judaic in roots, Orthodox Christian worship in Jerusalem is not that much different, when one starts to understand liturgical worship from which Judaic worship is. Most of the first Patriarchs of Jerusalem were converted Jews.
Disclaimer: I'm not Ethiopian, just stating historical facts, not doctrines.
If a Gentile being grafted into the tree (the tree first being Jew) if God pours rain down on that tree does the branch that is grafted in not quench its thirst also, of course it does. The Gentiles who turn to the Jewish Messiah are "adopted" making them spiritually Jew. A Gentile that converts to Messianic Judaism is a spiritual Jew by circumcision of the heart. The definition of convert is, to transform. Are we not transformed by Christ. I think what happens with this question is the confusion starts when we mix up Jew by blood and Jew by Jewish faith " Judaism". Judaism is not about blood because anyone can become a "Jewish believer" by adhereing to the laws of the Torah, so if a Gentile be grafted into Christ, the "Jewish Messiah" than you are now also a Jewish Messiah believing soul, right. So my answer is yes, when you choose to make Christ your Saviour you now are His bride, one body. You are what He is. Verses on this are: Ephesians 2:11-22, Romans 2:29.
Yes. The Messianics are by definition Christians, although they might not like being called that. The group is made up of 80 to 90% non Jews. It is absolutely possible for a non Jew to join Messianic Judaism. Ironically, this group will not allow you to do a Jewish conversion if you wanted to, and no sect of Judaism accepts Messianics as Judaism. It is accepted as true Judaism only by itself.