Annoy Squidward Day is a day that is mentioned in the episode "Employee of the Month" as part of a joke. It is first shown on a calendar.
It is supposedly held annually on February 15. Squidward is bothered, so when he is asked what day it is today, he answers, "Annoy Squidward Day?" SpongeBob laughs and says that "Annoy Squidward Day" is on this day, and points to the calendar.