I don't think they would like it..they like to scratch in the run..
I don't know about foxes.. we have coyotes. .the dog keeps them away.
I use mouse traps in coop to keep mice down. Set the traps in a small cage to keep chickens away from traps.. I don't have rats but you could do the same for them. . Make sure the cage has big enough holes for rats to get to traps.
You can easily make a welded wire 'apron' facing out all around the run to protect from large digging predators. To keep out foxes, the fence itself must not be some cheesy chicken wire, as they can bend and rip it. If the fence is chicken wire, laying down a hardware cloth or welded wire apron would be useless. To help keep mice at bay, a raised coop (even just five or six inches) is very helpful, and so is putting down bated traps where the chickens cant get to them. During the day, chickens will eat mice, but during the night mice will nibble on the tail feathers of chickens for protein and eat their feed.
That reminds me, MikeTheGardener, chickens can get their toe nails/toes caught in the hardware cloth while they are scratching around, and brake them (the toe) before they realize the toe is even caught. worms7- unless you put several inches (7-8) of dirt on top of the hardware cloth, pack it down, and plant some grass on top or throw down a thick layer of leaves to keep the chickens from getting to it, it poses a risk to your girls. Although broken toes can be fixed much more easily than some other injuries, I'd rather simply avoid having to if I can. Not to mention it is a waste of hardware cloth when you could just dig down and bury or made an apron around the run with it. Laying rocks on top of the apron or using tent staples to pin it to the ground helps prevent the digger from squeezing under it. You can even turn the foot or so of apron bottom into a flower bed around the run to hide it if you think it's ugly.
I think maybe if you at least put plywood (that can hold a chicken) over the wire it might be okay but I think it would be easer to just screw or nail the plywood or any other wood down in the first place.
If you put wood down in the run it will eventually rot. You would not want to use treated lumber either. Those hens spend a lot of time scratching and if there is anything under their they will eventually get down to it. That means splinters. You are not going to keep mice out because the smallest hardware cloth is only 1/2 inch and they will squeeze through that. Stick with the welded hardware cloth and keep it on the outside buried. I buried concrete blocks standing up that were 1 inch thick and 8 inches long. If something wanted to dig under they were going to have to go down 8 inches and then back up the other side.