Jim -- is there any evidence that a mormon missionary has left the mormon church and converted to Christianity while on his mission?
Yes, that has happened. One remarkable case was that of Jeff Pate (who is dead now). I met him fifteen or so years ago right after he came out of Mormonism. He called Ed Decker from the missionary apartment he was staying in and Ed actually drove to where he was, picked him up, and escorted him out of the apartment.
I have received mail from missionaries who have said that as a result of watching the Mormon Dilemma video they left the Church. Another wrote me to say that he went into a Christian bookstore to "check out the opposition" and began a process that led to his exit.
So those things happen. Obviously they don't happen frequently, but they do happen.