Are Messianic Jews Jewish?
No. A Jew who accepts Christianity might call himself a “Jewish Christian,”
but he is no longer a Jew. He can no longer even be counted as part of a
Conversion to another faith is an act of religious treason in Judaism. It is one of
the worst possible sins that a Jew can ever commit. Along with murder and
incest, it is one of the three cardinal sins which may not be violated even
under pain of death.  It’s a big deal.
Rabbi Moses Isserles demanded a formal conversion back to Judaism for those who converted out of Judaism but who then wanted to return to Judaism. (One who practices idolatry denies the whole of the Torah.)
He demanded ritual immersion (mikveh) and repentance before a court of three (beit din). You will see this also in other Responsa literature: Radbaz, Responsa III, 415; Moses Isserles to Yoreh Deah 268.12; and Hoffman, Melamed Leho-il II, 84.
Maimonides himself wrote that if a Jew converted to Christianity, he or she was no longer a Jew (Yad, loc. cit. 2:5.). Also see Maimonides, Hilchot Mamrim Perek 3, Halacha 1-3, as well as in Maimonides's Mishnah Torah, Avodat Kochavim 2:5.
Rabbi Moses Isserles demanded a formal conversion back to Judaism for those who converted out of Judaism but who then wanted to return to Judaism. He demanded ritual immersion (mikveh) and repentance before a court of three (beit din). You will see this also in other Responsa literature: Radbaz, Responsa III, 415; Moses Isserles to Yoreh Deah 268.12; and Hoffman, Melamed Leho-il II, 84.
Harries, Richard (August 2003). "Should Christians Try to Convert Jews?". After the evil: Christianity and Judaism in the shadow of the Holocaust. New York, NY: Oxford University Press. pp. g. 119. LCCN 2003-273342. ISBN 0199263132. "Thirdly, there is Jews for Jesus or, more generally, Messianic Judaism. This is a movement of people often of Jewish background who have come to believe Jesus is the expected Jewish messiah.…They often have congregations independent of other churches and specifically target Jews for conversion to their form of Christianity."
Kessler, Edward (2005). "Messianic Jews". in Edward Kessler and Neil Wenborn (eds.) (GoogleBooks). A dictionary of Jewish-Christian relations. Cambridge; New York: Cambridge University Press. pp. 292–293. LCCN 2005-012923. ISBN 9780521826921. OCLC 60340826. "From a mainstream Christian perspective Messianic Judaisms can also provoke hostility for misrepresenting Christianity."
Harris-Shapiro, Carol (1999). "Studying the Messianic Jews" (GoogleBooks). Messianic Judaism: a rabbi’s journey through religious change in America. Boston, MA: Beacon Press. pp. g. 3. LCCN 98-54864. ISBN 0807010405. OCLC 45729039. "And while many evangelical Churches are openly supportive of Messianic Judaism, they treat it as an ethnic church squarely within evangelical Christianity, rather than as a separate entity."
In the document ‘What Evangelical Christians should know about Messianic “jews”, Martin Meyer, founder of 'jews' FOR JESUS: reveals:
‘We are an arm of the local **Church**. We are Evangelists. We are accountable to the Church. As we win and ***convert*** Jewish people, we urge them to take their place in a local Evangelical **Church**, or establish a congregation. Our duty is to aid the **Church** at large and to gather in the Lost Sheep of the House of Israel.’
- from ‘Holy Terror’ by Conway and Seigelman
1) Yad, loc. cit. 2:5.
2) Pri Megadim, Eshel Avraham 55:4.
3) Sanhedrin 74a.
4) Sifri on Num. 15:22 and Deut. 11:28; Yad, Avodas Kochavim 2:4. Cf. Horios 8a.
5) Tshuvos Rivash 4, 11, Tshuvos Rabbi Yosef ben Lev 1:15.