Some folks are bad neighbors, no matter what and there is little you can say to them that will sink in unfortunately.I moved into my house a year ago, and this neighbor (we'll call her "Jane") moved in just months after, next door. I live on an acre and a half, and all of my chickens are secured in their run with wire of the top. I do this to protect them from predators, and from just wandering.
Jane's coop is obscured behind her house, while mine is visible from the road (my yard doesn't have many trees), so my concern is always that the chicken-less neighbors will think these are my chickens doing this, and call the county on me.
Any suggestions? I'm very confident just telling her to keep her chickens contained will not be well received at all.
Is your yard fenced? A small dog, especially something like a Yorkie or other varmint chasing breed, in a fenced yard will gleefully keep your property free-ranging chicken free.
I own both chickens and Yorkie. Chickens have learned to stay the heck out of my "no chickens allowed" fenced side yard since Dee Dee started patrolling it.
Rescue a Yorkie mix, (they make great companions and are fun, smart, loyal, little dogs that don't shed) teach it chicken patrolling and the neighbor will not be able to complain as long as your dog is kept inside your fenced yard.
I live in the country and even if someone complained to the county here they would just shrug and say "Not against the law". So I can't advise on that one.