They have accepted Christ as the Messiah, does that mean they are still Jewish?
Ethnically, yes. By rabbinical standards, no.
You have to remember that being Jewish is both a religious affiliation, an ethnicity, and a “culture” or “civilization”. This is how Shai Cherry, speaker in the Teaching Company’s lecture series, “Introduction to Judaism”, describes the term “Jewish”.
So, you can very well have an individual who is ethnically Jewish, but affiliates with a non-Jewish religion, such as Catholicism. There are even Jews who are unaffiliated with any religion, and are atheist or agnostic. You sometimes hear the term, “secular Jew”, referring to a person who has Jewish ethnicity, but does not practice any particular religion.
(Incidentally, this creates a lot of turmoil and angst for Gentiles seeking to convert to Judaism, as they cannot understand how it is acceptable for ethnic Jews who are non-observant of their faith can be considered “Jewish”, but converts who are quite zealous for Judaism and even religiously observant of Jewish law have to go through a very strenuous, difficult, and long conversion process. But alas, this is a different topic, and I digress.)
The Hebrew Catholic movement is in communion with Rome. Most are ethnically Jewish, but have spouses or family affiliations that are Catholic, and have therefore embraced Catholicism. The Mass is celebrated in Hebrew, and most Hebrew Catholics are located in Israel.
From what I have read thus far, they are permitted to retain and practice Jewish cultural (and biblical) traditions, such as keeping kosher, celebrating Passover, wearing tefillin (phylacteries), etc., while still remaining Catholic. The idea being that just as German Catholic and Polish Catholics, for example, have their own cultural identity and traditions, so do Jewish members of the Hebrew Catholic community.
Messianic Jews are not affiliated with Rome. There is also a large population of Messianic Jews in Israel, though they don’t receive the same press and publicity as other strands of Judaism. Messianic Jews do not celebrate traditional Catholic liturgical services nor sacraments as typically expressed in the Catholic faith. They believe in Christ as the Messiah, but they more closely follow an alternative version of Judaism than Catholicism. This raises, apparently, quite a bit of controversy among their Jewish brethren who do not accept Christ as the Messiah.