I would say it depends on the integrity of your chicken run. Mine is as safe as I believe I can make it. I leave the coop door open at night all summer, it gets way too hot here to "coop" them up tight in the summer heat and I don't have a problem. Plenty of clean, fresh water, food, I would...
Years ago, I kept button quail as pets. The adults mated, eggs hatched, but I had a similar experience. In my limited experience they are hard to brood in captivity. I would keep trying however, they make such wonderful companions.