Do Messianic Jews accept the entire New Testament as a base of their belief? Yes.
The Holy Bible is the main religious text for Messianic Jews. They follow the teachings of the Torah (Old Testament) while also holding the New Covenant (New Testament) as God’s Truth. They believe Old Testament prophesies about the coming Messiah were fulfilled in the New Testament with the birth, death, and resurrection of Jesus Christ.
- Bible - Messianic Jews believe the Bible is God’s Holy Word and follow the Old and New Testament. The Old Testament serves as a history of the Jewish nation, God’s covenant with Abraham, and the commandments given to Moses. The New Testament fulfills the Old Testament through the coming of Jesus Christ, the Promised Messiah and God’s Son.
- Jesus Christ - Messianic Jews believe that Jesus Christ is the Promised Messiah prophesied in the Old Testament. He is referred to in Hebrew as Yeshua and regarded as the way of salvation for all who believe in his death and resurrection. They believe that Jesus’ death was payment for the sins of all. (Acts 2)
- Trinity - Most Messianic Jews believe in the Trinity as God in three parts. God the Father, God the Son, and God the Holy Spirit are all displayed in the New Testament Scriptures through the Gospels (the first four books) and the book of Acts.
- Sin - Messianic Jews acknowledge that sin is any behavior that is against the teachings of the Bible. Sin can only be forgiven through faith in the death and resurrection of Jesus Christ. (2 Corinthians 5)
Most, but not all, Messianic Jews accept the orthodox view of the Trinity. God the Father, God the Son, and God the Holy Spirit as three representations of the same divinity.
God the Father: Messianic Jews believe in God, and that he is all-powerful, omnipresent, eternally existent outside of creation, and infinitely significant and benevolent. Some Messianic Jews affirm both the Shema and the Trinity, understanding the phrase "the LORD is One" to be referring to "a differentiated but singular deity" and "eternally existent in plural oneness".
God the Son: Most Messianic Jews consider Jesus to be the Messiah and divine as God the Son, in line with mainstream Christianity, and will even pray directly to him. Many also consider Jesus to be their "chief teacher and rabbi" whose life should be copied.
God the Holy Spirit: According to some Messianic Jews, the Spirit is introduced in the Old Testament, is the inspirer of prophets, and is the spirit of Truth described in the New Testament.
You say: “Stating «the Word was God» and «the Word became flesh» he sets the base for the thought that God, the Holy Spirit and Jesus are the same, which is not the Jewish (and Muslim) concept of God.”
Please be aware that the the Father, the Son and the Holy Spirit are not “the same”. That is not the Trinitarian concept. Trinitarians acknowledge that there is only the One Being of God within whom subsist the three personages of Father, Son and Holy Spirit. Each is co-eternal and co-equal but they are DISTINCT, one from the other. Source: Athanasian Creed.