We have a Jack Russell mix as well...
He is a.. well not the best mannered. But its not his fault. He was abused and not properly animal socialized. He is great with children and people in general. But awful with other animals. We have had him just over two years and he is almost 7 years. The day we got him, the little dummy lit into my 140# Rott. Our Rott was well socialized and he kinda looked funny at the little 12# "necklace" he was now wearing and rubbed it off. That Lil bugger did that four times... Each time my Rott edged up the repercussions. My Rott and I taught my Jack more "manors" that first week than in his entire life I think. As the owner of a Jack, you need to be on top of him. This breed needs to be very much under your thumb. He is not the best breed for someone not willing to do/be that. So much so last October when we got this 1 yr old Lhasa Apso he did not make a peep! He accepted her, with a couple growls and sniffs. He lightly lit into her when she came home from the groomers. But almost instantly realized he knew who she was a stopped instantly. But it was not an overly aggressive occurrence.
We recently made the leap and now have chickens. I was concerned about the Jack, but not our Lhasa Apso. The first day we scolded the Jack three times. He gets excited when they stretch and flap their wings, but has remained in control. Our Lhasa is perfect.. we call her the Chicken Watch dog... cause she sits and watches them!
As mentioned, its not his or not entirely his fault. Remember he is a ratter, bred to hunt. Anything that runs gets chased, its like a game a game that can turn deadly.
Have you thought of a muzzle when the dog in in the yard with the birds?
Good luck and I hope you keep your dog.
CW