Is "Messianic Judaism" considered to be a legitimate sect of Judaism? Could someone (gentile) convert to "Messianic Judaism"?
So-called "Messianic Judaism", meaning something that claims Jesus is the aforementioned messiah, is not accepted by mainstream Judaism. It is presumably possible to "convert" to their religion as their main mode of operation is to lure Jews into their churches, but that won't have any standing with most Jews. A conversion to Judaism not overseen by torah-observant Jews isn't valid, and by definition their leaders aren't observing the torah. If they "convert" a gentile he's therefore still a gentile, and if they "convert" a Jew he's still a Jew (who has gone off the path) (see here for example).
See this article from Aish HaTorah about these groups:
No matter how disconnected a Jew may be from Judaism, he is still likely to be appalled by the idea of worshipping Jesus. And that poses a great dilemma for Christian missionaries seeking to convert Jews.
Given this problem, some missionaries got the idea to try a backdoor tactic. They invented "Jews for Jesus," which uses a whole lexicon of Jewish-sounding buzz words in order to make Jesus more palatable to Jews. [...]
The sad thing is that tens of thousands of Jews (including an estimated 50,000 in Israel today) have fallen for this falsehood.
Ironically, Jews really could be called "Messianic Jews." One of Maimonides' classical "13 Principles of Faith" is: "I believe with complete faith in the coming of the Messiah, and even though he may delay, nevertheless I anticipate every day that he will come."
Aish HaTorah isn't making a p'sak ruling here, but the attitude in this article is consistent with what I've heard from many others too. I think you'd be hard-pressed to find Jews outside their communities who consider so-called "messianic Jews" to be practicing Judaism.
Individual Jews may accept you, but the State of Israel and Establishment Judaism in all its forms will not accept Jewish convert to Messianic Judaism as still being Jewish.
Like it or dislike it, it is what it is.
What is the basic difference in Christianity and Messianics? Not much
What is the basic difference in Christianity and Judaism? A lot
I would say if. Jew were to convert to Messianic "Judaism" it is just a stepping stone to Evangelical Christianity just as Karaisim is sometimes a stepping stone from Christianity to Judaism. Messianics are denying the 13 principles by believing the messiah has come not that he will come.