The holiday involves one having to do the opposite of what they normally do (exclusions being breathing, eating etc.)
Role in episode
Squidward is desperate to move out of Bikini Bottom after SpongeBob and Patrick throw a surprise birthday party for him the moment he wakes up even though it really is not his birthday. Later on, Squidward states that everything a good neighbor does, SpongeBob does the opposite, which gives Squidward an idea.
The following day, Squidward is acting cheerful and playing the drums loudly in front of SpongeBob's house. A confused SpongeBob wakes up and goes outside to ask Squidward why he is not acting like his grouchy self. The latter tells the former that it is Opposite Day and explains that on this holiday people do the opposite of what they normally do. SpongeBob then goes back inside and his happy-go-lucky personality transitions into grumpiness and anger.
He then hears a knock on the door and realizes that it is his best friend Patrick. SpongeBob tells him to go away and that he never wants to see him again. Shocked, Patrick bursts into tears as a result of SpongeBob's betrayal of him. However, SpongeBob then tells Patrick that he did not actually mean it and only said it in celebration of Opposite Day. Patrick has never heard of the holiday, so SpongeBob shows him the ropes.
SpongeBob later starts copying Squidward and Patrick joins in too. Once Patty Rechid comes to take a look at Squidward's house, SpongeBob explains all its flaws and Patrick (disguised as Squidward) pops through a painting of Squidward's and Patty wants to know who he is. SpongeBob, going by Squidward's name, tells him that the guy is Squidward. Patty is left confused, wondering how there could two Squidwards at once.
After rebuilding SpongeBob's house, the actual Squidward walks into his own home and informs her that he is the real Squidward and that SpongeBob and Patrick are just impersonating him.
After Gary, in disguise as Squidward, comes into his house, an outraged Patty refuses to sell his house out of confusion over who is the true Squidward, much to Squidward's disappointment.