I live on an acre of fenced land in a very rural area. When I first moved in a neighbor's dog killed three of my cats in my yard because there was no gate to keep her out. I built a gate. Four or five years ago I started raising chickens and have had a LOT of trouble with raccoons (gates and fences don't bother them). They do love chicken dinner! My cats watch the chickens from a safe distance, but get out of the way if a chicken comes at them. All of my cats were rescued, unwanted kittens and I have had all of them neutered or spayed and get their shots regularaly (raccoons carry rabies).
Most cats will not bother your chickens, but there are humane ways to keep they away if you are worried. It is NOT necessary to kill them. If your neighbor won't keep his/her cat at home, tell them that they will have to pay for any chicken it kills, and they will think twice about letting them into your yard. Most urban areas (and some rural areas) require cats as well as dogs to be "restrained" so you should check with your local animal control . Please don't kill an animal that is doing what its instincts require. Blame the irresponsible owner and try to deal with them.