Thaddeus is seen wearing the hat and claims to wear it because he is bald. When Mr. Krabs flies out of the Krusty Krab, Thaddeus bows his head and then takes off his hat to take a bite. The Hatty Patty was also the idea of Squidward, and Mr. Krabs makes it.
His name is an homage to Thaddäus Tentakel, the German name for Squidward. The German dub subsequently changed his name to Fischward to avoid confusion.