Yes, a mink is absolutely in the weasel family and they, just like the entire weasel family are merciless killers. If they get in your coop at night, they will kill every bird in it. If they can. People always say that animals kill only what they can eat. Horse apples!! Just about every predator that preys on chickens will kill many times more than they could ever possibly eat.
The only predator I've ever heard of that was illegal to catch or kill was raptors (birds of prey). I've surely never heard of any jurisdiction where it was illegal to trap, shoot or otherwise stop any predator like weasels, coons, possums or foxes that's killing your stock. Even your neighbor's dogs that are allowed to run loose are fair game even in the most animal-friendly areas. This nation was founded and remained an agrarian society up until very recent history. Laws protecting your stock have been on the books for generations.
The only predator I've ever heard of that was illegal to catch or kill was raptors (birds of prey). I've surely never heard of any jurisdiction where it was illegal to trap, shoot or otherwise stop any predator like weasels, coons, possums or foxes that's killing your stock. Even your neighbor's dogs that are allowed to run loose are fair game even in the most animal-friendly areas. This nation was founded and remained an agrarian society up until very recent history. Laws protecting your stock have been on the books for generations.