Breezly isn't thrown out of Camp Frostbite at all during this episode. He leaves on his own at the end of the episode, also ending the series.
Breezly: Always leave them laughing!
Breezly: Too bad you don't have the measles, Sneezly. This medical book tells us how to cure that.
Breezly: (to Sneezly) You'll feel like a new seal, when we're finished.
Breezly: We will disguise ourselves in snowballs.
Sneezly: We look more like a couple of oddballs.
Breezly: (to Sneezly, after he has sneezed and destroyed their igloo) It's the third time this week your sneeze has unglued our igloo!