This.That's only legal if the dog is on your property. You can't shoot your neighbor's dog if it's properly contained on its owner's property and a chicken happens to wander into its territory.
You can't shoot a dog (or any animal for that matter) because it killed an animal that doesn't belong on its property. It's the owner's responsibility to keep their animals contained. If they fail to do that, the consequences are on them, not on whoever happens to find the animal first (especially if the animal is somewhere it shouldn't be, like on a neighbor's property, who happens to own a dog...)