On the one hand rubber horse stall mats are fine (if rather pricey) as coop flooring, under your bedding.
On the other hand they are NOT predatorproof. Rats chew through them pretty easily, and although I have not actually seen in action the type configuration you're describing, I strongly suspect that a dog or other digging predator would just bull its way up through, pushing at the stall mats til they push up enough to come out from under the run fence at the side, letting the predator in. Also, having a solid floor like that to your run will prevent the chickens from scratching and digging, which is a really major part of their life and a shame to deprive them of IMO.
I would suggest a wire (or very-large-paver) apron, as the best alternative to digging in buried wire. And leave the run floor be earth.
Good luck, have fun,
Pat