IF you can prove the neighbor's dog came into your yard and attacked your dog, the neighbor is liable for all documented harm.
If you did not originally notify the police and animal control of this attack, do so now.
Compile all vet bills, prescriptions, etc., make copies and send the neighbor a demand letter for the total damages, with copies of all invoices. Send it by certified mail/return receipt.
If the neighbor fails to pay, your next option is small claims court.
The foregoing is for general information purposes and does not establish an attorney-client relationship.
Yes, in most jurisdictions your neighbor would be liable. Report it to animal control or the governing local authority. You will need to prove that the dog came into your yard. Your neighbor's homeowner's insurance should cover this. They may attempt to limit the damages to your dog's fair market value.
This is not to be construed as legal advice. I do not have an attorney client relationship with you.
Missouri has some interesting dog laws:
273.030. Dogs may be killed, when
If any person shall discover any dog or dogs in the act of killing, wounding or chasing sheep in any portion of this state, or shall discover any dog or dogs under such circumstances as to satisfactorily show that such dog or dogs has or have been recently engaged in killing or chasing sheep or other domestic animal or animals, such person is authorized to immediately pursue and kill such dog or dogs; provided, however, that such dog or dogs shall not be killed in any enclosure belonging to or being in lawful possession of the owner of such dog or dogs.
273.036. Dog bite without provocation--owner liable for damages
1. The owner or possessor of any dog that bites, without provocation, any person while such person is on public property, or lawfully on private property, including the property of the owner or possessor of the dog, is strictly liable for damages suffered by persons bitten, regardless of the former viciousness of the dog or the owner's or possessor's knowledge of such viciousness. Owners and possessors of dogs shall also be strictly liable for any damage to property or livestock proximately caused by their dogs. If it is determined that the damaged party had fault in the incident, any damages owed by the owner or possessor of the biting dog shall be reduced by the same percentage that the damaged party's fault contributed to the incident. The provisions of this section shall not apply to dogs killing or maiming sheep or other domestic animals under section 273.020.
2. Any person who is held liable under the provisions of subsection 1 of this section shall pay a fine not exceeding one thousand dollars. The remedies provided by this section are in addition to and cumulative with any other remedy provided by statute or common law.
You may give me a call if you have further questions.
My comments are general in nature, are not legal advice as to your specific issue, and do not establish an attorney-client relationship. Disregard this solicitation if you have already engaged a lawyer in connection with the legal matter referred to in this solicitation. You may wish to consult your lawyer or another lawyer instead of me. The exact nature of your legal situation will depend on many facts not known to me at this time. You should understand that the advice and information in this solicitation is general and that your own situation may vary. This statement is required by rule of the Supreme Court of Missouri.