My 2007 Eclipse has an irritating issue. When I let off the break pedal I hear it meow at me. It's been like this for years, but it irks me. I've heard various theories about air being trapped in various places in the brake lines (and have had my brakes bled as remediation, which didn't help), but was hoping someone with the mechanical chops could provide a bit more comprehensive list of things to check.
I have dealt with several brake squeak problems. Since it's just a squeak, I would bet it has nothing to do with trapped air. First, be mindful of your personal safety before starting any inspection or repair. If you must get under the car, ensure you have adequately supported the vehicle on jack stands before moving any part of yourself under the car. Do not be tempted to "take a quick look" with the car only partially supported by the jack, since you may be further tempted to perform the repair without first securing your safety. This can lead to leaving the car supported only by the jack for an extended period, greatly increasing the risk of jack failure and serious injury. That said, I would suggest you look at these as possibilities:
Check if it's just your pedal squeaking by trying the brake with the car off and in park. If the sound is from inside the cabin, and it doesn't go away even when the parking brake is engaged, it may be the pedal. If it does go away, or the sound it coming from outside the cabin, the pedal is likely not the problem.
Old or Worn Out Brake Line
I once had a faint squeak upon braking (in this case both while pressing and releasing) on a 1999 Toyota Corolla. I ignored it for a while, which turned out to be a big mistake. Eventually the brake line from the master cylinder to the rear drums ruptured, draining almost all of the hydraulic fluid. Luckily for me, this happened in my driveway after I had set the parking brake, so the mechanical brake held the car in place. I ended up having the car towed to shop where they installed all new metal brake lines. Call the shop and ask first if they can and will replace the brake lines, to avoid paying for multiple tows.
If this is your problem, you may (or may not) be able to see signs of stress on the brake line under the car. On my Corolla, there was supposed to be a plastic shield in place under the brake line to protect it from road debris, but it had long since cracked and fallen off. A replacement cover was not available, so I figured I'd hope the new steel lines would hold up to whatever punishment came up during the car's remaining useful life.
Improper Brake Cable Tension
If your car has disc brakes on the rear wheels, the problem is likely not brake tension, as they should not need manual adjustment.
First, eliminate the pedal as the cause of the squeak. Set the parking brake and see if the squeak persists. If it does not, it may be that the brake cable is too loose.
This recently happened on my 2003 Toyota Corolla, and adjusting the brake cable tension was a simple and painless fix. This video shows the process for the Corolla, which should be similar enough to other cars with drum brakes.
Essentially, you lift both rear wheels from the ground and see how far they spin with a modest push. Usually, 1.5 revolutions is about right. If it spins too far, the cable is too loose. Too little rotation, and either the tension is too high or you forgot to release the parking brake. To adjust the tension, remove the tension screw cover gasket (if you are lucky enough to still have them) from the rear of the brake assembly. Then insert a flat screwdriver into the opening and press it against the teeth of the adjuster. Don't overdo it by pushing the adjuster only one or two detents ("clicks") each time, checking the rotation of the wheel in between. Adjust one wheel at a time, and check the squeak before and after adjusting the second wheel. When satisfied, replace the cover gaskets, and return the wheels to the ground.
My guess is that you're hearing vibration or chatter of the pads against the rotor as they move away when you release the pedal. You might start troubleshooting this by finding a sloping parking lot or other place where it would be safe to have somebody outside the car listening as you reproduce the sound. Let the car drift down the hill and apply then release the brake. Ask your helper to identify the wheel(s) that the sound is coming from. S/he may need to move around the car to be certain that they've got the source right.
With that information in hand, jack up the offending wheel and see if you can spin it fast enough by hand to recreate the sound – if you can you're golden as it will be quite easy to test to see if you've found a fix.
Next pull the wheel and tire and carefully inspect the caliper and its guide pins and sliding surfaces. See if you can still reproduce the sound by spinning the rotor. If you can then you may be able to identify the location either by ear or by touching parts to damp out the sound.
The two most likely possibilities seem to be that the caliper or the pads may be hanging up and slowing the retreat of the pads when you release the brakes or that the "anti-squeal" spring behind the pads is vibrating and causing the sound.
Be very careful if you apply any lubricants that they don't (and can't) get onto the pads or rotors. If it is the pads vibrating against the spring or calipers you might try getting some "anti-squeal compound" which is flexible glue-like stuff that damps out vibrations.