The back squat is usually taught first by the Bulgarians as a means of building leg strength and getting a feel for that really DEEP position that you'll constantly go to after you get it. They really stress a deep back squat, so if you don't have one, I think it's pretty important.
And for your front squat having 60lbs restin on your fingertips, you still should have the bar on your shoulders. Make sure to get your elbows HIGH like in the videos and pictures I'm sure you've seen. Fingertips or not you should be able to feel the bar resting on your shoulders. You can front squat without your hands like bodybuilders do/used to because you don't really hold the bar up there, your shoulder rack does.
Work on the stretches for your wrists asap!