"I won't stay longer than I can help" or "longer than I can't help"?
No, one shouldn't.
If you can help it, you won't do it.
"I won't stay longer than I can help."
Maybe better is
"I won't stay longer than I can manage."
meaning
"I will not stay longer if I don't need to"
or alternatively
"I will stay longer if I do need to."