Should Christians celebrate Christmas?
I sympathize with those who want to be rigorously and distinctly Christian, who want to be disentangled from the world and any pagan roots that might lie beneath our celebration of Christmas, but I don't go that route on this matter because I think there comes a point where the roots are so far gone that the present meaning doesn't carry the pagan connotation anymore. I'm more concerned about a new paganism that gets layered on top of Christian holidays.
Here's the example I use: All language has roots somewhere. Most of our days of the week—if not all—grew out of pagan names too. So should we stop using the word "Sunday" because it may have related to the worship of the sun once upon a time? In modern English "Sunday" doesn't carry that connotation, and that's the very nature of language. In a sense, holidays are like chronological language.
Christmas now means that we mark, in Christian ways, the birth of Jesus Christ. I think the birth, death and resurrection of Christ are the most important events in human history. Not to mark them in some way, by way of special celebration, would be folly it seems to me.
I remember I lived next door to somebody back in seminary who didn't celebrate birthdays for their kid. The idea was, partly, that all days were special for their kid. But if all days are special then it probably means that there are no special days. Yet some things are so good and precious—like anniversaries, birthdays, and even deaths—that they are worthy of being marked. How much more the birth and death of Jesus Christ!
It's really worth the risk, even if the date of December 25 was chosen because of its proximity to some kind of pagan festival. Let's just take it, sanctify it, and make the most of it, because Christ is worthy of being celebrated in his birth.
There is no point in choosing any other date. It won't work.