You should do other things besides just squatting. You just can't push them hard. I would do some type of upper body press, maybe dips too, keep doing plenty of ab work.
I wouldn't pull from the floor, but that's just me. If I were do do this program and keep cleaning, I might work from the high hang. I'd most likely get my pulling from chins and rowing though.
During the unloading phases, you're advised to lift explosively, so I would clean then.
Remember this is a cycle to build your slow, raw strength.
If you run it, I'd love to see a journal, I toyed with the idea a couple months ago of doing it, but I didn't want to put the cleans on the back burner.