OK, didn't think this was relevant, but all but one of the birds died on its back. The one we watched died on its side, rolling over as it died.
Reading around, this is apparently a rare position for a bird to die in. Some heart issues, flip over disease and asphyxiation are indicated. Flip over doesn't make sense, I don't think, since these are primarily older birds and aren't fast growing, nor are they broilers. Asphyxiation might make sense if one got hold of something, but I can't imagine that many would, especially those that were confined at the time of death, though my husband's co-worker said he had a similar issue and it turned out to be a snake. Heart issue? They aren't all the same breed, but most came from the same batch at the same feed store. Except the hen who died in the nest box and the little chick that was runty and deformed from the beginning.
Unless it is some combination of external stressors putting pressure on the heart? I don't know. Why would a chicken have a heart attack while just sitting there?