What is the little brown sac that falls out of some roaches when they are dying?
Ans: Boomom, That is a sac of eggs, called an ootheca. See the pictures at URLs: 1) Blattinae, the Periplaneta family, with protruding and laid oothecae 2) German cockroaches, some females with ootheca. The females carry the ootheca protruding from their bursa which is at their tail end until, depending on the species, it hatches, they hide it somewhere or until they retract it into their uterus to incubate it until the eggs hatch. The German cockroach carries it around until it hatches and might be the culprit you are specifically asking about. The American cockroach carries it around for a day or two until it hardens and darkens and then hides it in a crevasse covered with some camouflage they glue in place with saliva. The ootheca is only gently held in the bursa and could easily fall off when the roach is roughly handled or dying.