Both of the guys who got sick from eating Beaver 'seemed' fine until they scraped the fat off of their pelts.
The first guy processed the carcass and waited too long till he got around to scraping the hide for the last bit of fat, so it 'appeared' like he was careless.
The second guy carefully smoked the meat that he could not get to, and he took the extra step to smoke the hide of the beaver with the fat, so I thought he'd be ok. I was really surprised that he also got sick.
I recall from a previous season that someone got sick from eating a muskrat, another critter that lives on the water so 'maybe' there is a connection.