There is no known origin for this joke. According to Urban Dictionary, it's a rhyme-arific and obnoxious way to respond to the obnoxious & ubiquitous question, "Guess What?" A much easier response than actually trying to guess the topic about to be brought up.
Here's some more:
Guess Where?
Answer: Chicken hair
Guess Why?
Answer Chicken thigh
Guess Who?
Answer Chicken pooh
Guess the rest?
Answer Chicken Breàst
See Adam Introvert's research on the potential origins of "Chicken Bùtt".
Check out 50 Terrible, Quick Jokes That’ll Get You A Laugh On Demand, featured at ThoughtCatalog.com.