It will depend upon state laws if you will be found liable for the medical vet bills of your neighbor's dog that you hit and thus if your insurance company would pay for the injuries of the dog.
If there are leash laws and your neighbor had their dog off the leash and it ran into the path of the car then you may not be held liable, again your state's laws, including negligence laws, will help decide if you should be responsible to pay the vet bills or not.
Most states, cities and/or townships have a leash law. With this type of law if a dog is not on a leash it is declared to be "running at large." In general if your dog is not on a leash and is "running at large," you are responsible for your dog.
In many areas of the US you as the driver would not be 100 percent liable since the pet owner has some legal responsibility to control their pet. Depending upon state negligence laws the driver may only be found to have contributory negligence (in the form of not having proper look out and striking the dog) since the neighbor's dog should have been under the control of a person and ideally on a leash.
All this being said, to find out about your state's negligence laws and if your auto insurance would pay the neighbor's vet bills you can contact your insurance agent. Medical expenses for an animal are not included in the policy language of most auto insurance policies. You may also want to see about getting legal advice or try contacting your state's insurance regulator. Their consumer division should be able to give you advice on your situation.