It's no surprise that most people think of Hanukkah as the Jewish alternative to Christmas. We tend to experience them in American culture as an either or--"Jews celebrate Hanukkah because we/they don't believe Jesus is the Messiah." I suppose this is indeed the way most Americans, whether Jewish or not, experience Christmas/Hanukkah.
It is, however, needless. Hanukkah, the Feast of Dedication or the Festival of Lights, was around before Jesus was. It celebrates the rededication of the temple in 164BC after it was desecrated by the minions of Antiochus Epiphanes IV (Dan. 11:31) in the Maccabean crisis. The story has of course grown over time. In the later form of the story in the Talmud, oil that would normally only last one day miraculously lasts eight days. This part of the story isn't in the books of 1 and 2 Maccabees, which date over 500 years earlier than the Talmud.
From a Christian perspective, this story is in the Old Testament period. That makes it no different in theory than the story of Esther or of Passover or of Jonah. Do Christians believe that God can perform miracles and did for Israel? Indeed, Jesus attends this festival in John 10:32, and Hebrews 11:35 seems to allude to the story of 7 martyred brothers during the Maccabean crisis.
So why do some Christians have an adverse reaction to Hanukkah?
1. Because it has become a religious boundary line in later Jew-Christian cultural marking. It's an artificial line historically, but real as a socially constructed reality.
2. Because this story is in the "Catholic" books, 1 and 2 Maccabees. Come on guys, the Reformation's been over for 500 years. Get over yourselves. As far as history goes, which is not at all the touchstone of theological truth, but nevertheless, as far as history goes, there's more historical evidence for this rededication than for the overwhelming majority of stories in the Old Testament!
3. Ignorance of history
4. Can't think of any more.
I'm not suggesting that Christians should start celebrating Hanukkah. But I believe we can if we want to.