The reason Messianic Jews are a special problem, is because they lie. Most would say they are practicing a form of Judaism, when in fact they are clearly Christian by religion. They have misappropriated things from Jewish religion and culture in an effort to lure in unsuspecting Jews, to whom they lie and say that "Jesus will make you more fully Jewish."
The reason Christian missionaries to the Jews is a problem should be obvious. Jews who become Christians, in addition to engaging in avodah zarah which is forbidden us (idolatry), are basically assimilated into the Gentile world. Even if the initial convert continues to say he is a Jew, by the time you get to his grandchildren, they no longer identify as Jews. His entire line is lost to Israel. Thus, these conversions to Christianity are threat to the Jewish people as a whole, to our survival.