I've seen my dark Cornishes scoop them against their necks tightly hold them under what would be a chin, if chickens had chins. Sort of like when they stretch out and tuck them underneath themselves, but they hold tighter so they can pick them up. I have found hens, when I picked them up, to be holding an egg up under a wing, or between their legs.
If my Cornies see eggs in the next nest when they're broody, it just drives them nuts that they don't have them. You can watch them plotting to take the eggs. It's pretty funny. They stretch waaay over, and pull the eggs as close as they can, then they start trying to get them from one nest to the other. I usually come and take them, before they get them moved, because I remove the new, unmarked eggs daily anyway. Otherwise they'd add new eggs everyday, and end up with a mess of partly developed chicks, when the originals hatch out.
Oh yeah, mark the eggs. Sharpie, pencil, or crayon. Sharpie stays best. Very important. So you know if there are new eggs, which ones to remove.