The objection that Christians ought not to celebrate or observe Christmas is usually based on four reasons. 1. It has pagan origins and therefore Christians ought not to observe it. 2. it is a Roman Catholic tradition that involves the Catholic Mass and therefore Protestants shouldn't be involved in it; 3. it violates the Reformed doctrine of the Regulative Principle of Worship; 4. it involves images and statues of Christ as a baby. Which is a violation of one of the Ten Commandments against images of God. Since most professing Christians are, in principle, Trinitarians, Christmas involves images and statues of Christ who is God in the flesh.
I'll answer objections 2-4 quickly, because the main purpose of this blogpost is to address objection 1. Since this issue is:
a.) being used by cultists to lure people away from Christianity, and
b.) by non-Christians (e.g. some atheists) as a case of hypocrisy on the part of Christians.
While I'm a Calvinistic and Continuationist Baptist, and therefore loosely "Reformed", I don't currently hold to the Reformed doctrine of the Regulative Principle of Worship. The reasons for that are beyond the scope of this blogpost. Regarding the claim that Christmas or Christmass is Roman Catholic, I would simply say that when Protestants observe Christmas they usually don't do so in conjunction with the Catholic Mass or other Catholic approaches to worship. So, Protestants can stripe the holiday of Catholic worship. Similar to how most Christians who use the names of the days of the week aren't worshipping Thor on Thursday, or Woden/Odin on Wednesday, or Saturn on Saturday [etc.]. When it comes to the objection that it involves images of Christ, I don't believe it is inherently sinful to have images of Jesus since He was a historical person with flesh, bone and blood. To deny that Jesus can be portrayed in art is to tacitly (usually unintentionally) deny the incarnation. That the Son of God really and truly entered space and time. I don't see a problem in celebrating Christmas so long as it's understood [or at least not denied] that no one really knows how Jesus looked like in the flesh and therefore the images of Christ are artistic suppositions/speculations. And so long as the images aren't being used as aids in worship. While I'm often uncomfortable with images and statues of Christ, I think they are permissible for the stated reason that the incarnation teaches Jesus was not only truly God, but also truly man. A true historical man can be portrayed in art. To deny that Jesus can be portrayed in art is to deny His true humanity.
With that said, here are videos that address the claim that Christmas is pagan.
Christmas is Not Pagan PARTS 1 & 2 by InspiringPhilosophy [Michael Jones]