Well Doglover, looks like nobody has had this exact problem to be able to pass on some personal experience. I confess that "styes" is not something I have ever come across as well. Not a Britt thing, but rather a dog thing.
Did a little research, not much there either, but from what you report it seems that there is a recurring infection either from bacteria or possibly a fungus of some kind. It comes and goes so the body is working at getting rid of it....that is what the stye is, a corraling of the bad stuff and trying to expel it from the body. I think the vet needs to take a smear of the goop that is in the stye and see if s/he can determine the origin, bacterial, fungal, maybe even viral, dunno. They may or may not be able to do that test in-house, and it might even be expensive, no clue.
What I would do, while you are trying to find out what is the underlying reason for the problem, would be to get a simple eye wash/drops solution that can be used frequently during the day, especially after a visit to the out-of-doors. And then first thing each morning and last thing each night before bedtime. Similiasan (brand name) for allergy eye or dry eye is a good brand and the drops can be used often or as required. This will help keep the eyes rinsed out, especially if the cause is from something environmental, that is, from outside the body.
You might also get on a program of immune system support since this may be a symptom of a systemic imbalance. A good probiotic, such as Acidophilus, preferably in powder form, added to at least one meal a day would be beneficial. Also, a small raw garlic clove crushed and added to one meal a day would be beneficial. If you use the garlic it is suggested that after 2-3 weeks that you begin to pulse it, meaning give it every 2-3 days rather than every day. These are proactive and will do no harm.
I am assuming that the vets have ruled out "cherry eye", ingrown eyelashes, and turned-under eye lids. You seem to have made a connection with his hunting habits....being an "earth dog" is not all that unusual for a Britt, or any canine hunter

Good luck....I hope you will keep us in the loop about what you find out and if you are able to solve the problem. We can all learn from your experience.