i'm gonna say shocks... if it happens when you're going over bumps... it doesn't have anything to do with brakes. Check your p/s fluid level, and maybe do a flush and re-fill on your power steering. It might be time to give it some fresh fluid. Also, make sure the squeeking over bumps isn't your hatch, thats a huge problem with hatch back integras. You can fix it by adjusting the rubber stoppers or putting something around the U bolt latch.
A few noises like that shouldn't ruin the experience of driving your teg. You should hear my car, everything creaks, moans, squeeks... thats what i love about it, good ol' urethane bushings and all my chassis stiffening. Not to mention removal of sound/vibration dampening material and noiser tires
did you try opening your hatch, and just bouncing the car a bit to see if you can't provoke the sound from coming out? If its the shocks... you will definately hear it that way, if its something else thats caused by a more abrupt jarring motion, then it might not make a noise.. but at least you'll know if its the shocks or not.
i have a similar problem. i traced it to the upper control arm things attached above the strut and coil (they have two bolts, and is shaped like a triangle..i think its called upper control arm). for some reason they squeak when they bounce up and down with the shocks and suspension and especially seem to squeak more when its cold. at first i thought it was the dash but i even took apart everything and manually closed it togethor it made the noise. i even went to a junkyard looking for new ones and when i came across an integra it had the same problem with it. im not sure what to do at this point...by brand new ones probably