Brakes have loud high pitched squeal noise

Those shims are actually there to stop brake squeals. The pads are going to vibrate, and there is nothing we can do to stop that, but there are a number of things we can do to reduce the noise we can hear. Brake squeals are caused by the vibrations of the pads being transmitted into the piston, then the caliper housing, where the sound is amplified. The shims present one more obstacle the sound has to travel through.

There are two things to consider with brake noise. One is to reduce what is developed in the first place, and the second is to reduce how much of it we hear. The first trick to reduce what is developed, when you install new pads, use a flat file to cut the sharp leading and trailing edges of the friction material. Even dragging those edges on a concrete floor can be sufficient. That removes the "fingernails-on-the-blackboard" screeching. We used to do this on a bench grinder, but that is no longer recommended because it takes away the squeegee action after driving through deep water. That water causes one type of brake fade. I found that if you can eliminate that screeching during the break-in period, the pads will not cause that noise for the rest of their lives.

Next, run the flat file over the piston's contact area where it rests against the pad's backing plate. You do not need to shine it up. You are just insuring there is no high spots of rust or dirt. The pad needs to sit firmly against the piston. Run the file over the caliper's outer fingers where they contact the outer pad's backing plate, again, to insure there is no dirt stuck between them and the pad. Just before you install the pads, coat the backing plates with special high-temperature brake grease. This is the important step all professionals do, but most do-it-yourselfers are not aware of. "Rusty Lube" is one trade name I am familiar with, but there are many others that are just as good. They all have molybdenum disulphide in them. The grease will not travel like petroleum-based greases will, and it will not smoke or burn away. The pads are going to vibrate, and the grease here will allow them to slide over the piston and the outer fingers on the caliper with less transmission of the vibrations to the caliper. Put some grease on the outer pad too. You do not have to coat the entire backing plates, just where they rest against something.

The caliper mounting hardware must be inspected for rust pits, bent bolts, and things like that. Replace any rusty parts. Do not bother trying to clean them with a wire brush because if the chrome plating has already lifted, rust will continue to form there. Those rust spots will impair the caliper's ability to slide back and forth. That can keep one pad slightly applied all the time, and under light braking, that caliper may not apply as much as the other one. All the mounting hardware and slides related to the caliper sliding must be coated with the brake grease.

One of the biggest offenders for causing noise is grease contamination of the friction surfaces, meaning the rotor and the pads. I have run into a few managers who demand their employees throw away new pads and shoes if they get grease on them, but it is sufficient to wash and scrub that grease off with Brake Parts Cleaner as long as that is done before those parts go through their first heat-up cycle. The linings and cast iron rotors and drums are porous. Once heated up, grease contamination will soak in, then you will never get it out and you will always have a brake squeal.

Professionals are even careful to not pick up rotors, drums, and linings by their friction surfaces to avoid getting fingerprint grease on them. If your hands are reasonably clean, your fingerprints are not likely to cause a problem, but it is a chance we would rather avoid.

The worn shims you found will not cause noise, at least not directly. Screeching and squealing noises have to be caused by something that is moving. The fact they are worn through proves something has been moving and rubbing on them. Typically that is caused either by the pads vibrating and there was no brake grease on them, or they simply corroded away over time. Brake grease is not meant to stop that corrosion, but it will slow it down.