in my situation this was not sufficient. in order to rehab from ankle injury. i would say the minimal effective weight would have been about body weight plus about 1/2 body weight. and much more safety came around bodyweight plus body weight on bar. also it was a weighted calf raise, with range of motion with ankle starting lower than ball of foot, that gave me back my ankle. after simply body weight squats etc did not.
an issue to consider will be- when you apply force with one foot , how much force do you apply?
in order to run , how much force will you need to apply with an individual foot or side?
when you are applying force with both feet/legs etc at the same time, how much force does each side apply?
the basic problem, and the failure of the body weight squat is-
when you are running all your body weight goes onto one foot/side. (and when you pass your foot your heel goes lower than the ball of your foot and there is likely some element of plyometric at least some times in your stride)
the jump from 1/2 body weight to full body weight is too big of a jump.
in order to progress this jump gradually it is easiest to add weight to activities done on both feet.
until one foot/side is able to lift the whole body weight.
while you might not need to squat with an addition of 1 x body weight, there are other motions
that you will need access to in order to run. many issues for example relating to just the first 1/2 inch of ankle lift.( others relating to impact forces especially while speeding up or slowing down.)
that a body weight squat, or even a 2 x body weight squat etc, will not protect you for.
basic situation being. there are multiple injuries. that if you were to need to rehab through, that it will make sense to lift at least 1/2 your body weight , and closer to your full body weight, before its "safe" to start using light light dumbbells. for example 2 to 5lb, on one foot.
"easiest" way to think about it. and i guarantee near every program gets this wrong, is think about the force nec for one foot and the force prepared by activities done on two feet, and bridge this difference gradually by adding weight to activities done with two feet planted. to regain the power to use each individual side.