I see a lot of chicken lovers on this thread, but you have to step back and view this from 30,000 feet. You cannot have it both ways. If the chickens are livestock, then they are not pets - they are property. Personal feelings about the chickens are moot. If, on the other hand, the chickens are pets, then they are not protected under this law. If a dog breaks into my yard and kills one of my dogs, I cannot shoot it.
So this dog was shot for destruction of property, alive or not. Would you shoot a dog if it got into your yard and tore up your garden as your first response? I'll assume that you are not so sentimentally attached to your broccoli. Livestock = property not pets. Therefore shooting the dog because it killed your beloved chickens is revenge, not protection. This isn't to say that I don't have affection for my chickens that is on par with my rabbits, quail, dogs, cats and other misc. beasties - but in such recognize that someone else might have a similar (or greater) level of affection for their pet. Rather than getting revenge by shooting the dog, get compensation and help educate the neighbor. I hope that the lawsuit doesn't succeed because that might work to limit the ability for people to protect their livestock, but in this case such ability was obviously abused.