Where I live, it's illegal to allow fowl to roam at large. And if they cause any damage to the neighbors, I can be taken to court, and I can be made to pay for their damges. Also where I live, I've been told by the Sherif that I have the right to protect my property, and if an animal (any animal) that has been allowed to roam at large, damages my property, I can shoot it. We don't have animal control here. I have a few choices, but amoung them is to shoot them.
I don't know a place in the U.S. where one doesn't have the right to protect their property in one way or another, and if you live in a rural zone, your neighbor most likely has the right to shoot any predators, including chickens. Remember, a predator doesn't have to be a dog, or a coyote. It can be anything that is causing harm or damage. And one has to admit, chickens sure are destructive.
You can call your Sherif if you can't find the information online to see what your rights are - and his.
However, I would do as you plan: put up that fence or make a big run for them to range in and don't listen to him. He sounds like a troublemaker from what you describe. Just cover your butt and do as you planned. Just make sure you can build a fence, as others have suggested.
For others who commented: I can think of many reasons why a neighbor would not want chickens on their property, among them is that they poop everywhere and they destroy your gardens and landscaping. For those who have put time and money into their gardens and landscaping, our wandering chickens cause damage to personal property. That said, everyone has the right to protect their property.
It may not be a big deal to us because we own chickens and we have modified our landscapes and gardens to prevent the chickens from destroying them - or we just tolerate them destroying stuff. Our property, our choice, right? But our neighbors are not expected to tolerate our animals, anymore than we have to tolerate theres. It goes both ways. And the law is on their side.
I understand that some of you think it's horrendous to shoot a chicken, dog, cat, buffalo, whatever. But the fact is, we all have the right to defend what's ours and it should be our right. As silly as it sounds to protect your porch from chicken poo (it smells, it's corrosive and our neighbors shouldn't have to drag a hose out every time our chickens poop on their porch), it's just as silly to them that we want to protect our "expendable" chickens. We can argue about our individual methods of protecting what's ours till we run out of breath and keel over, the fact is, the choice is ours.
If we all treated each other like we want to be treated, the pests and predators section would be very quiet.