I finally found what was causing my brakes to squeak. I thought for a long time it was the front brakes, but it turns out to be the rear brakes! They'd squeak when not applied and more so when cold.

The rear pads have a 1/16" shim attached to the back of the pads held on with one rivet that holds the piston clip on. This shim was rotating and dragging on the rotor flange causing the squeak. I just ground that overhanging section off and we'll see if it returns. The noise was driving me nuts and making people stare at me thinking something was wrong. ALL the brake hardware was new; calipers, rotors, pads, hoses, brackets BOTH front and rear. I even did the Akebono swap. It turns out I didn't need ANY of that stuff! I'd saved my old pads and after checking them, this was happening every time I changed the pads. Quiet for a while, then squeaking. If you have this problem, check it out!