yes, it depends on your area, but most states do allow you to shoot a dog that is ATTACKING livestock. That means that you have to wait until the dog is physically going for your birds. Other states allow you to shoot if the dog is harassing your birds, completely depends on your area.
What state are you in? PA, for example, lets you shoot if the dog is attacking. If the dog is just caught in the coop/run, all you can do is confine or chain the dog and call animal control or the police to come and get the dog. From the looks of it, in PA you aren't entitled to compensation if the birds are free-ranging?
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reimbursement for damage to a domestic animal by a dog, whether or not the domestic animal is directly damaged by the dog or is necessarily destroyed due to damage caused by the dog, if all of the following apply:
1. The damage occurs when the domestic animal is confined in a field or other enclosure adequate for confinement of such animal. (See footnote 3 for case that describes enclosure requirement).
2. The damage was not caused by a dog owner or harbored by the owner of such damaged domestic animal.
3. The owner of the offending dog is unknown
ETA: if many states, you have to actually catch the dog in the act of attacking, you can't do it if you come and find dead birds in your yard.
just do a search for "dog attack livestock <your state>" you can also get advice from your local animal control and police department.
Also, be sure that it is legal for you to discharge a firearm on your property. I know that many people here have chickens in urban areas where shooting might be illegal
