"Get a lawyer involved and let him do the talking for you."
Lawyers will tell you that any settlement will be limited to damage amounts you can prove.
Vet bills, for instance would be reasonable. OTOH any thought of "hurting" them by some huge monetary lawsuit is not likely in the cards.
They will need to hire a lawyer of their own, which will hurt plenty! The plaintiff's lawyer can also sue for legal costs in addition to direct costs, including time off from work, etc. A claim for mental anguish arising from the neighbors' negligence is also worth pursuing, IMHO. It need never go to trial; the key is to hit 'em hard and extract a settlement through their throat.