We have a Priefert, Arrowquip and Titan chute. Only the Arrowquip has rubber on the floor, the Titan has a breast bar (I don't love the Priefert, but it does the job). I rarely have a cow or calf go down in any of them and when it happens, it's usually after an unpleasant procedure like treating a hoof, lancing an abscess, etc.
@Fred nailed it.
Don't get all cowboy in the corral and rush 'em in the chute. As far as squeezing, I'd say it's case by case. With a one 'n done shot, quite often I don't even use the headgate or the squeeze.