Hi Mike, it's likely too hot inside for them, even with the misters and shade. Your coop needs more ventilation. Can you remove one entire wall from it and replace it with hardware cloth? Also, certain kinds of parasites, for example poultry ticks, will make them reluctant to go into the coop at night. Check the roost and the crevices around it within a half hour after sunset tonight to see if they are there. I really suspect the lack of ventilation to be the issue, but it's always worth ruling out ticks here in AZ.
ETA, it is o.k. to leave them out, but your enclosure has to be very predator proof.