Now, curiously enough, I've never heard anyone actually object to this. I've never objected to it myself. What I have heard, pretty much every Christmas, is Christians objecting to people saying "happy holidays" -- including, last Christmas, a woman I know to be quite religious yelling very rudely at a younger woman who had uttered the offending words to a decidedly mixed group of people.
The legitimacy of the Christians' possessiveness about the holiday is in any case tenuous. Christmas is most likely a mere adaptation of Saturnalia, the pagan Roman festival of gift-giving and revelry celebrated in late December, in an effort to make Christianity more palatable to the pagans by merely changing the pretext for their most popular holiday rather than abolishing it. Other associated customs such as the Christmas tree are probably of pagan Germanic origin. No element of modern Christmas -- not even the claimed association of December 25 with the birth of Jesus -- has any basis in the New Testament. I rather doubt there's a Biblical passage in which Jesus instructs his followers to get snotty with people who say something as innocuous as "happy holidays", either.
Nevertheless, I am more than willing to concede that Christmas today, regardless of its history, should indeed be regarded as a Christian holiday. After all, considering what it has become -- all the crass consumerism, mob scenes, greed, squabbling, stress, and those godawful "carols"* -- who would want it back from them? They broke it, they own it.
I just wish they'd refrain from taking out their understandable frustration with all those shopping-mall lines on people who use greetings they disapprove of.
*The only Christmas music I like is "Winter Wonderland", which someone once told me isn't even a "carol", and the Mannheim Steamroller version of "Good King Wenceslas", which I'm sure would never be played in any church.