When I was in the car business a long time ago, there was a guy who came by with laundry and carpets every week. I slipped him a twenty for a walkway carpet with the rubber backing, and it has lasted me 20+ years and still going strong. Using little squares sounds a bit crazy to me and kinda amateur looking. A nice advantage to the carpet is you can put down duct tape markers at each stand foot, and be set up quickly the same every time.
A full carpet versus little squares lets your weight and the weight of the whole kit keep the pieces that want to travel in place. It also marks your territory so to speak on the stage. So yeah, rubber or gel backing, pile on top. You don't want to be the guy that scratches up the floor at a venue, right?