The officer that Jess is confronted by has a badge that says "Nicholas," referring to the fact that Santa is also known as Saint Nicholas.
When Nick gets caught (literally) with his pants down, he yells "This is my nightmare!" He said the same thing in the previous Christmas episode when he was trapped with Jess and Paul who were breaking up.
Nick (Jake Johnson) at one point says "Nick Miller; turning lemonade into lemons since 1981" implying that he was born in 1981. Jake Johnson was actually born in 1978