If all the eggs from six chickens have been disappearing for weeks it could be a family of rodents. Young opossums can get into very small holes. I would suggest that you empty out the coop, determine were the predator is entering and then seal any holes and cracks and maybe you even need to tighten up your doors.
But as far as letting your chickens out during the day, what ever is taking your eggs has has plenty of opportunity but hasn't harmed your hens. so whatever it is, it probably can't.
I wanted to add that we don't put any food into the coop. We do make sure the hens are out from sun up till sun down and since they are always up on the perch until I open the coop in the morning and don't rush to the food or water (so far) it's working for us but that might not work for everyone.