Brake squeal and now steering wheel shimmy

I can suggest a few things. First, it sounds like the right caliper is sticking and not releasing properly. That will cause the pads to grab harder in one spot on the rotor once per wheel revolution. That may or may not be felt in the pedal and / or the steering wheel depending if there is thickness variation warpage or if the braking surface is simply tilted relative to the center hub. If you HAD to use a c-clamp to retract the piston in that caliper to fit the new pads in, that piston is sticking. They should move freely enough to be pried in with a screwdriver before the caliper is unbolted.

When a caliper sticks, the brake will stay applied and overheat and melt the glue holding the lining material together. That will lead to glazing of the rotor and braking material. Glazing leads to squealing. To verify that is the cause, remove the wheel and use course sandpaper to scuff up both sides of the rotor to remove the shiny surface. That roughness will grind the layer of glazing from the brake pads too, then the squeal will be gone for a little while.

Next, check for rust or scale between the rotor and mounting hub. You can see the effects of that by jacking the front end up, running it in gear, and watching the wheel to see if it's moving back and forth sideways as it rotates. This is more common when reinstalling used rotors and the normal round rust spots aren't scraped off, but scale can break loose and get stuck in there too.

It's also common for new brake rotors to warp, especially those made in China. A simple machining will take care of that. Cast iron parts made here are "aged" for 90 days before their final machining. Parts made in China are perfectly fine but they cast, machine, pack, and ship them right away, then they age on your car and sometimes warp. One machining is all that's needed to solve the shimmy or vibration.

It's normal to hear the pads making a light noise when you're right by the wheel. Also, since you could press the pistons into the calipers by hand, they are not sticking and very likely glazing is not the issue. Check the caliper mounting bolts to be sure they aren't rusty or bent. The caliper has to be able to slide back and forth sideways freely.

As for the rust I'm referring to, if you look at the hub when the rotor is removed, you will see the wheel studs installed through holes in the hub and there will usually be one or three additional holes between the studs. Water splashes up there, through the holes, and onto the backside of the rotor. That's where the round spots of rust build up. Those have to be scrapped off to insure it will sit perfectly straight on the brake lathe and on the hub. If the rotor is reinstalled in a different orientation with that rust spot still trapped in there, it will prevent the rotor from sitting perfectly parallel to that hub. That will cause the caliper to shake sideways back and forth with each wheel revolution, AND it will make the wheel and tire wobble too.

That said, that isn't the concern here because you installed new rotors so there aren't going to be any rust spots.

The screws you drilled out sound like they are the ones that hold the rotor to the hub. Those are used to hold them on while the car is being flipped around on the assembly line. You don't have to put them back because the wheel will hold the rotor tight.

Getting back to the noise concern, there are many things professionals do to prevent noises and there are some things do-it-yourselfers can do to cause noises. Here's a copy / paste version of a reply I post quite often:

To prevent a crunching sound when cornering, put a light coating of high-temperature brake grease between the hub to rotor contact points. This is especially important on older GM fwd cars. Do not get any grease on the pad or rotor friction surfaces. That includes fingerprint grease. Some very picky shops and mechanics will discard pads that get soiled with any kind of grease, but it is usually sufficient to wash all friction surfaces with brake parts cleaner. If this is done before final assembly, there should be no problem. If that grease is there when the parts get hot from normal braking, the grease will soak into the linings and the porous cast iron rotors. It will cause a squeal and never come out.

Another way to prevent squealing brakes is to remove the sharp edge from the leading surfaces of the linings. I used to use a bench grinder, then switched to a flat file. Now I've found it is sufficient to rub the sharp edges on the concrete floor. That removes the "fingernails on the blackboard" screeching. It seems if you can prevent that squeal during the break-in period, they won't squeal later either.

Run a flat file over the pistons and caliper fingers that contact the outer pads. You don't have to shine those surfaces up. The goal is just to be sure there's no dirt or rust that will prevent flat, even contact between the pads' backing plates and the pistons and calipers. Uneven contact will allow the pads to vibrate more than normal. That can set up an audible squeal. Those same contact points should also be coated with brake grease. That can let the pads vibrate without transferring the noise to the calipers where it will be amplified.

Anyplace the pad backing plates or calipers rest on a metal mounting bracket should also have a light coating of grease. That includes chrome-plated mounting bolts that hold the calipers to the mounts. If those bolts have rust pits or are bent, they should be replaced.

Yes, it can easily cause a shimmy, but that is due to a sticking piston which it appears you don't have. That type of shimmy gets worse as the brake heats up.

It's normal for the pads to stay in contact with the rotor, but just lightly. That creates the squeegee action that clears off water to prevent brake fade after driving through deep water. When you apply the brake pedal, the seal around the piston sticks to it and deforms, (bends) a little. When you release the brake pedal, that seal straightens out and pulls the piston back just a couple thousandths of an inch. That's what releases the pressure between the pads and the rotor.

Remove the rotor and check behind it and the hub for foreign materials lodged in between. If any hard rust had gotten in between the hub flange and rotor, the hub would not seat laterally.