During the first break segment, the Skeleton Crew is playing a version of "Let's Have a Patrick Swayze Christmas" from the original MST3K's ("Santa Claus Conquers the Martians").
Tom says "Stop squeezing that tinsel!" when a Mr. Whipple is addressed in the movie. This is a joke about the Charmin toilet paper commercial where Mr. Whipple tells people not to squeeze the Charmin.
This episode was not originally part of the initial Kickstarter pitch. The response to the Kickstarter was so positive, the entire season run (and all the planned bonus episodes) funded in full before the campaign wrapped. Since there was still plenty of time left for further funding, Joel decided to add a "Christmas Episode" in the last few hours of the campaign.
The stinger was Santa learning how to do his 'ho-ho-ho' laugh.