I agree with the person who suggested electrical fencing. I think that's the most sensible solution.
As for the person who thinks that anyone who cares about their cat will keep it indoors - we used to keep all our cats indoors. Then we got Rowdy. The second time he jumped off the 2nd story patio to get out and came back limping, we decided that he REALLY wants to be an indoor/outdoor cat. He generally sleeps cuddled up with us at night. He goes out the cat/dog door and comes back, at will. Sometimes he'll rush at the chickens, then stops, but mostly he ignores them. Our other cat completely ignores them.
If I thought any of the neighbor cats were a problem (attacking the chickens rather than just curious), electric fencing would be my first option, since it would also be useful against other predators. The suggestion to get a dog is also a good one. I don't recommend shooting the cat. It won't keep other predators out. And I have to say, if someone shot my cat, they'd have a lot more to worry about than one cat.