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.
The Messianic Jewish Alliance of America
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:
Concerning the acceptance of non-Jews into their congregations ("Messianic Gentiles"), Kesher Journal reports (emphasis mine):
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,
The Union of Messianic Jewish Congregations
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:
To again quote Kesher Journal:
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 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.
1 Cor 7
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.