The usual advice is not to run in rocker bottom shoes, but if works for you, go for it. Just be real careful not to twist your ankle, because they have poor lateral stability.
As far as squats, it's not good to be on the toes of one foot. It twists your pelvis and spine, which could potentially injure your back. Try squatting with a plate or board under both heels (bodybuilders do this). Or maybe squat in the rocker shoes?