Boxes, for nest boxes? No, they're really not going to work.
Nest boxes need to be private, and they need to have bedding/shavings, etc to make the hen feel cozy and cushion the eggs from breaking. Keeping the bedding in the box means some sort of lip or facing on the front bottom edge, or the birds will just spill the bedding out on the ground while trying to arrange things to their liking. You've seen your dog try to "make" his bed? Chickens do the same thing with a nest, and will enthusiastically empty a nest box unless you design it to hold the bedding inside.
They'd also need a perch or some such, to be able to access the upper boxes. They can walk into the bottom one, but they'd have a hard time flying/flapping up to the level of the higher ones and physically getting into the box.
In my main coop, I use plastic totes for nest boxes. They're deep, I can put the bedding in and they can't scratch it out, the hens can hunker down and feel private. One downside is, a heavy hen perching on the edge to get in or out can tip the tote over, but usually it just rights itself when she jumps down. I do occasionally have a tote on it's side, but the bedding prevents eggs from being broken. One could probably use a screw and washer to fix the tote to the wall to keep it from tipping, but I also like that I can take them out and hose them if needed.
