Although they probably won't "Bust" I wouldn't recommend lowering the car that much before getting new ones. OH and definitly get a camber kit like Mogg said
Yah, definetely get new struts. The stock struts werent meant to handle the abuse of a lowered car. And camber kit is a must when lowering the car any height if you want to save money on tires.
last time i check integras didnt have shocks??? sturt is the proper term i believe, but ya stock struts suck ass no matter what it will make ur suspension bounce worse with aftermarket springs on there. get some new monroe struts there cheap and work out well, unless ur pockets deep then go for kyb adjustables or koni's.