I'm in Virginia. If someone else's animal comes on my property, I am allowed to call animal control. Or I am allowed to pen up the animal, and charge the owner for taking care of it. But it's still not my animal. (Yes, various details are spelled out in the law, about kinds of animals and how much to charge.)
I am not allowed to keep the animal.
I am allowed to kill someone else's animal only under specific circumstances (like if it's a dog actively chasing livestock, or if any animal is endangering a person.)
Many other states have similar laws.
In the OP's case, I think a chicken-proof fence (cheap netting on cheap poles) could be a good solution.
Or catching/trapping the chickens and taking them to the animal shelter.
Or reporting the code violations.
But I think that keeping or eating the neighbor's chickens should not happen unless the neighbor agrees. (Could ask: "If those chickens come here again, can I eat them?" If the neighbor says, "Go ahead," then it's fine.)