Not sure about mink, but weasels will kill everything in the pen. The telltale sign is dismembering all or most of the carcasses. They can get in anywhere their head will fit, usually through the recall funnel. If accessing through the funnel, they might not find their way back out so will furiously chew the inside of the pen at a seam. After the carnage, they will hide or crawl in any dark place inside the pen if they can't get back out. Easy to trap out of the pen with a cardboard box with a small folding door.
Solid, varmint guard doors tightly hinged over the funnel entrance works very well as long as it is closed each evening even if some of the birds are not back inside. If so, open it again in the morning.
A solid guard door that is effectively latched will work against weasels and clever raccoons when used to cover the pigeon "bob" door on a homing coop. With no solid guard door, raccoons will back out of the pen and rip birds right through the bobs.
3-strand, heavy wire, electric fence with a good energizer works like magic on black bears bent on destroying your coop to get the grain. Use a cutoff knife switch mounted head high to quickly cut power to access the pens. Keep weeds off the lower strand.
Solid guard doors are not for everyone because of the dedication involved to close them each evening after use but lose a hundred birds and you might think differently.