Lubricating the pivot screws should help solve the squeak.
Part of the problem with Strat tremolos, is that there is quite a lot of side-to-side movement because the screws don't hold the trem tight onto the body, the trem pivots on them, balanced by the strings and the springs, so it never returns to quite the same place every time. This is the most important factor in tuning stability.
There are two things you can do to balance the trem:
1. Balance the springs - if you are using three springs, run the two outside springs from the outside edge of the trem block, to the two 'inside' claws next to the middle spring, in a 'V' shape. If you are using 11s, and want the trem to rest flat on the body, you could probably use four springs, in which case, use the outside holes in the trem block and the corresponding claws, leaving the middle one free. This will help control any lateral movement of the bridge.
2. Set the 6 pivot screws like this: The two outside screws should be almost flat on the tremolo base-plate. The best way to judge this properly is to take off the strings and springs, allowing the trem to rest flat on the body, then tighten each of the two outside screws in turn until the back of the trem raises off the body, then loosen them off a touch until it rests flat again. the four middle screws should be raised ever so slightly, so there is a tiny gap between the screw-head and the base-plate, about the thickness of a couple of sheets of paper. This reduces friction by stopping the trem binding against the screw heads, so it will return to it's original position better. Don't worry, it won't fall off like this.
Check your nut, make shure it doesn't pinch the strings. You can test it by bending behind the nut. If the string doesn't return to proper pitch, widen the nut slot.
Make shure the strings are properly wound around the tuning pegs. No more than three turns, or switch to locking tuners.
