A "yegg" is a slang term for a burglar or safecracker.
Mel Blanc based the voice and character of the Easter Rabbit on the equally morose Happy Postman he played on "The Burns and Allen Show".
Bugs calls the Easter eggs technicolor hen fruit.
The "dead end kid" Bugs attempts to deliver an egg to is a reference to the Dead End Kids, a group of young, rambunctious actors who made several films at Warner Brothers with famous leads, including Angels with Dirty Faces with James Cagney. Due to their often destructive antics on the Warner Brothers lot, their contract was terminated in 1939.
The hammer and watch trick would also be used in the TV series Hogan's Heroes, by magician-con man, Corporal Newkirk.