Is merely shaking hands with a Christian while he is reciting his prayer considered apostasy that expels one from the religion, and is it permissible for a Muslim to intend to re-enter Islam?

It is permissible to accept an invitation from a Christian for dinner, just as the Prophet, peace and blessings be upon him, accepted the invitation of a Jew. As for him holding your hand to recite his prayer, if it does not contain polytheism (shirk (definition)), or if it contains polytheism but you are not convinced by it, then that does not take you out of the fold of Islam. You must be wary of his desire to influence you, and you should strive to invite him to Islam through dialogue, seeking help from experienced callers to Islam, lending him useful books and tapes, and showing him Islamic websites. As for your intention to re-enter Islam, you should avoid it for fear of obsessive thoughts (waswasah). It is desirable to renew one's faith continuously by frequently uttering the word of monotheism (Shahada), performing good deeds, and asking Allah for guidance.