Do Jewish people celebrate Christmas like Christians?
What do they do on Christmas?
First, it should be noted that you are asking a very broad question, as, unfortunately, there certainly are many Jews who celebrate it as a secular, cultural holiday, and many Christians who classify themselves as Jews.
Mainstream Judaism, however, rejects Jesus. Adamantly, decisively, and without qualification. We do not believe he was a prophet, a leader, or any sort of deity. In fact, the things that we do believe about him range from some rejecting the idea that he even existed (at least in the sense that Christians believe - one person with one message), to the idea that he was a bastard (in the biblical sense, a son of an illicit relationship), to his being a rabble-rouser (in a negative sense), to his being a false prophet, all the way to his being a practitioner of dark magic and a purely evil person.
So, depending how extreme you want to get, no, Jews do not celebrate Christmas for any or all of the reasons above.
Some things we do on Christmas (this list ranges from the serious to the silly, so forgive me):
Abstain from studying Torah the night of December 24th, because Torah study strengthens spiritual forces in the world, even evil forces, and the evil forces are at their strongest on that night (this is over-simplified for the sake of this answer - more here).
Work extra hours to take advantage of a quiet office, and/or to swap a holiday with a non-Jewish colleague so as to have time off for a Jewish holiday.
Participate in organized community mitzvah projects (like this one).
Go to the movies.
Eat Chinese food.
Watch "Christmas time for the Jews" over and over again because it's pretty darn funny.
No we do not. We do not consider him a deity and therefore we have nothing to celebrate on that day. Some [mainly Chassidim] have the custom to abstain from learning the Torah on that day so as not to give strength to the Evil forces that are around on that day. Read more here.