I fail to understand why you think because a dog is on your deer hunting land you have the right to kill it. It is illegal.
Illegal for what reason? I'm not about shooting stray dogs. We had a dog come onto our lease last year that actually looked pretty bad. We kept him around for a while and fed him until he took off again. My issue is people should respect other people's land (leased or owned) and not encroach on that by trespassing or allowing their animals to roam free on it. It's all about respect.
How do you know that the dog was let out of the fenced yard by mail man? or crook breaking into the house? How big would the big bad hunter feel if he found out that the family lost all the christmas gifts due to theft and someone shot their pet?
