I use bins with a simple light diffuser panel (bought at Home Depot) cut to fit laid in the bottom and put a little water in the bottom of the tub. 12 clutches this year and with 3 down no issues other than the normal roll out failed egg.
I use this method, but I put a layer of moist vermiculite on the bottom and put the egg crate on top of that. Then put the eggs on the egg crate. I did it this way mostly so the water wouldn't slosh around.