I've tracked down a potential folk etymology of "butt that won't quit" from the phrase 'legs that won't quit', but I can't attest that in a dictionary anywhere.
My question comes from my perceived notion it meant 'Won't give up' with the 'ass' being a Synecdoche for the person in question, that is it's not just the ass that won't give up but the whole person. Perhaps the connection of Asses (the animal) being stubborn contributed.
I'm not the only one who thought this. I'm a native speaker though, so I didn't think it was language barrier. Maybe I am a naive speaker though.
The main reason I am asking is that I am wanting to use this in a novel to imply somebody who won't give up no matter what (I'm not after alternatives though), and wanted something punny like:
They had an ass that wouldn't quit. In fact <the main character> suspected it had take up a second job, and worked in a soup kitchen on weekends.
But that doesn't connote their stubbornness if this is an uncommon interpretation of the phrase.
Does this meaning work for that phrase at all, and if so how common is it?