He often says things like:
* "Eh... what's up, doc?"
* "Of course, you realize, this means war!"
* "That's all folks!"
However, the phrase about holding hands and eyes falling out isn't a specific line from Bugs Bunny. It's more of a funny play on words that could be attributed to him due to his mischievous and comedic nature.
It's possible the phrase is a mix-up or a humorous interpretation of Bugs Bunny's antics. He often plays tricks on other characters, so the idea of him causing someone's eyes to fall out from holding his hand is a funny exaggeration of his mischievous tendencies.