I would fight to keep my chickens.
Does the town have meetings? Go to one and ask to see the town ordinances on chickens. My cousin had to do that because she wanted chickens and the town kept saying it was against the ordinances. When they got to see the ordinances they found that they were allowed to keep chickens as long as they were housed in a "proper coop" meaning a roof over the run so chickens couldn't fly out and she is only allowed one rooster, and only if it doesn't crow too loud.
So request to see copies of the ordinances. If there are provisions in there for chicken keeping, your neighbor will have to let it alone.