I am very Puritanical and so I do not celebrate Christmas. These are my arguments taken from Gillespie’s English Popish Ceremonies and David Calderwood’s Perth assembly:
Holy Days-
1. Diminish the importance of the Lord's day.
2. Monuments of past by idolatry.
3. Idols themselves.
4. God commands that we work on common days and if there is no prohibition to work, such as the Sabbath, we are commanded to work and redeem the time (Exo 20:9, Eph 5:16).
5. Duet 12:29-32; Holy days such as Christmas are not simply traditional days for public remembrance, heritage or civil celebration but are synchristic acts of religious worship that combine elements of Christianity and paganism.
6. Not just a family day, but more solemn and religious than the weekly Sabbath.
7. The Lord’s Day is given to remember all of Christ’s work and has apostolic authority in it’s practice. No other day commemorating any other work of Christ is commanded or practiced by apostolic example and therefore forbidden.
8. Relics of idolatry are never to be Christianized as the Roman religion seeks. Therefore, our relationship to Christmas trees is at least analogous to what Hezekiah did to the brazen serpent that had of late been used in superstitious worship (2 Kings 18:4). He did not baptize it, he destroyed it.
9. Christmas DayThe day Jesus died is referred to as the Passover in John 18:39. This celebration begins after sundown on the 14th day of Nisan which on our calendar would be in the middle of March. According to Daniel 9:24-27 there are 70 weeks (Weeks are periods of seven years) determined for Israel. There are 69 weeks from the building of the temple to Messiah. The Messiah’s ministry is in the midst of the 70th week and an in the middle of this week he is killed and his death puts an end to the sacrificial system (Mat 27:51). Therefore, his ministry began at the end of the 69th week and continued until the middle of the 70th week, making his ministry exactly 3 and ½ years in duration. Jesus began his ministry when he was 30 (Luk 3:23) years old as was tradition among the Levite priests from the command of God in the law (Num 4:3). Therefore, his death was in mid March; 6 months before mid March is mid September and three years before that is obviously mid September. He began his ministry the same time as his 30th birthday, therefore he was born in the month of September not even close to December 25th . 10. Jesus was born and in the same day was lying in a manger (Luke 2:16). After eight days he received circumcision (vs 21). Even as Mat 2:1 and the context reveal, the magi who came bearing gifts came much later after his birthday; as is added by Mat 2:11 they found him in a house as compared with his manger birth. These were not birthday gifts and it was not the first Christmas celebration.
In reference to PF’s complaint about the synagogue this is a classic Federal Vision argument. I did quite a large study on the issue. You can read my article on it here:
The Synagogue and the Regulative Principle by Drake Shelton - The Kings Parlor
Not all things done in the Old Covenant need scriptural evidence from the Hebrew text that underlies our Protestant canons. Greg Price in his Defending the Reformation Regulative Principle of Worship; or, Was Synagogue Worship Regulated By God's Revealed Word? says, "a. Blood sacrifices were required by God, though no explicit authorization was recorded (Gen. 4:1-7). Thus, it must be inferred that God revealed His will concerning blood sacrifices to Adam, Eve, Cain and Abel, but did not record His original authorization in Scripture. b. Clean animals were offered in sacrifice by Noah rather than unclean animals (Gen. 8:20-21). Where does God specifically authorize clean animals and forbid unclean animals in sacrifice? Or where does God identify which animals are clean and which are unclean prior to the Levitical law?" etc.