There is an old adage that simply states "form follows function". In the world of farm barns and buildings, that notion applies like no other.
The point being that a barn of the type the OP was using.......while sturdy and useful for a lot of things......like sheltering large animal livestock and their feed.....or perhaps even some farm equipment and/or farm tools.......is not the best place for housing chickens. Not even the box stalls. I have two such stalls in my horse barn and predators can access them with ease. Same with the stalls in my daughter's horse barn.....same with just about any other barn I've been around.
In days past, on just about every farm that ever existed.....at least one that raised chickens.......the chickens were kept in special purpose built housing......chicken houses. Chicken houses provided what the barns could not.......light, ventilation, and most important of all, security.
Case in point, the OP found a dig site in the dirt......made by what they believe to have been a mink....but in truth, could probably have been made by a variety of varmints. The birds roosting in the rafters would have been low hanging fruit for coons, possums, weasel, a mink, bobcat, etc. All of which can climb into the rafters with ease. So while they were sheltered from the weather, they were not safe there from predators. A barn of that type would also be a haven for rats.
I think one of the hardest lesson folks have to learn when they start raising chickens is the danger these birds face on a constant daily basis. We don't know what we don't know. Regretably, we often find out the hard way when we are faced with the carnage dished out from a whole variety of predators.........who tend to be very good at surviving in the wild on things far harder to catch and kill than our chickens, which to predators.......are about as tough to snag as sitting ducks. The loss can be both traumatic and overwhelming.
Back to the problem at hand, to keep birds safe in a confined area of a livestock barn, you basically have to build a jail cell or chicken house within the barn. The box stall may have four solid walls (bars on doors don't count), but you also need to secure the floor and ceiling as well. Best floor is cement......but that is a big project and big expense to go with it. A person could line the entire floor with hardware cloth to stop diggers (and that includes rats), then cover with litter. The ceiling has to be equally secure......something like 1" x 2" welded wire, supported by rafters itself, and secure all the way to the corners.....no gaps anywhere. In short, a fully enclosed cube. The kicker being the chickens ain't going to like being confined to this cube. They would prefer the rafters........but they don't know what we know......which is a chicken roosting in the rafters of a barn is dead already.
Option B is to consider moving them out of the barn in purpose built chicken house that is secure and meets the other needs of the birds as well.
Probably not what you wanted to hear........but.............