Thinking on the other side of the situation, which I doubt pertains to the OP's situation since the neighbor doesn't seem to care, how would you feel if you had a pet, that was contained appropriately and somehow accidently got out. How upsetting to find out that the person that found your dog kills him instead of returning it or calling you to pick it up when it was an honest accident.
If my dogs got out, even by "accident" and went after a neighbor's chicken while their child was in the yard?? If that kid went running after their chickens to save them while my dog was attacking the chicken....YES I would want them to shoot my dog. The value of a human life is far greater than a dog's. People love to think their dogs are incapable of hurting a child, but when push comes to shove, dogs are not wired the same as humans...period.
Granted, our dogs are all 10 lbs or under, but are still perfectly capable of killing a chicken and hurting a child. All three are trained in obedience and two of them participate in Agility and Rally. They are extremely well-behaved dogs and have never so much as snapped at anyone. I still wouldn't ever trust them around chickens, especially with a child in the yard.
(I fully respect your point of view and your opinion, I just disagree with it. It's a good thing we're all different. Otherwise, the world would be a very boring place!)

To the OP:
I would definitely PUSH the issue of monetary reimbursement as well as contacting the animal control in your area and find out what (if any) leash laws there may be. If you terminate the dog if it comes on your property again without taking any other action; your careless neighbors will most likely just get another dog and the cycle will repeat.
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