Here's my update on this. I dumped one garbage bag full of tiny-shredded paper into the nesting boxes (4). The next day, two chickens had laid their eggs on the pine chips on the coop floor, probably because they had no clue what the paper was all about. I put those eggs into the nesting boxes, and the other hens laid just fine. The eggs look very clean and nice, though we have one jerk-chicken who likes to roost on the side of one box and poop into it which dirties the eggs laid in that box, but that's not related to the paper.
A few days later, the first chicken of the day to lay once again laid on the floor of the coop. I moved her egg into the nesting box and left it there for the day, and the others followed suit and laid in the nesting box. So there might be a bit of a training curve, and a dummy egg wouldn't hurt. That's my experience with it.
Haven't tried shoveling it out of the boxes to change it yet, so I may hate it when it's time for that, but we will see.