Do I have any recourse against his and his actions?
Many states have statutes specifically authorizing the destruction of any dog off its property and which is harassing humans, livestock or domestic animals. If your state has such a statute - and Utah, a farming/ranching state almost certainly does - your neighbor's actions were almost certainly lawful. Indeed, you may be liable for the damage your dog caused your neighbor.
If your dog posed no such threat and was simply shot because of what he MIGHT have done, the statute, assuming there is one, would probably not apply.
You would do well to have an attorney assess your rights, if any, and liability, which is likely, on these facts.
Trademark Application Attorney
I agree with my colleague - any suit will be your word against your neighbor's as to whether the dog was actually bothering his chickens and children and if the judge believes his version of events, then he was within his rights to shoot your dog.
You should speak with a local attorney for further guidance. The Utah Bar Association can give you a referral: http://www.utahbar.org/