My chickens favourite nest box is a cat carrier box without the door, just set in the corner on the floor. Before that they just had cardboard boxes. So yes I think your litter trays will be fine (and easy to wash). The problem with cardboard is that red mite can hide in the corrugations, so need replacing frequently. I think they like the cat carrier because it has light coming in through the vents in the side, but they feel safe and secure.
Just noticed you said in the run, should be ok as long as its a calm, dry space near to the coop. Oh and use a fake egg to give them the idea.