I just recently picked up a Schecter Damien with a Floyd Rose. First FR I've ever owned and so far I'm pretty happy with it. The only thing I don't understand is when a guy like Joe Satriani, for lack of a better term, "pulls on it," and it makes that signature FR squeal.

When I tried it the bridge bottomed out before it got to the squeal and I'm not sure where to go from here. Could the bridge be too high? User error? The guitar came with an extra spring but I'm not sure that will help.

Sorry if this is a stupid question but I'm just trying to clear this up.
"When the going gets tough, the tough start drinking"
That squeal is most likely a natural or pinch harmonic combined with the whammy.

Extra spring won't help; it'll just make it harder to manipulate the whammy.