We have dogs running properties and sometimes even game here ,and it's a real problem sometimes. It's not illegal to shoot dogs here if they're damaging your property , injuring/killing livestock , or menacing/attacking people . It's not legal to "hunt" a dog in any state that I can think of unless you count coyotes .

Short of fencing the place off the poster may not have much recoursesince he's already been told not to shoot them . At best he'll have to file more complaints until he has enough baggage to take the yokels to court . At worst he may have to fall back on the 3 S rule . Tough choice either way .