For the shifting to one side during the squat problem, it may be related to an inactive glute.
Only having one glute that does not fire correctly is not uncommon at all.
In this case all the unilateral work in the world won't help unless its activating that glute--however, you should still be doing unilateral work to even out other imbalances.
Buy Cressey and Robertson's DVD here in the store. It should help you figure out if your glute is firing or not, and if it is, well you still have an awesome DVD to help with lots of other stuff.
And by the way, none of this is to contradict the previous posts. They were all great ideas too, and just actively concentrating on doing it right will help you cue the right pattern.