This has been a problem since the advent of semi-metallic brake pads. It is inherent to their nature to setup harmonic oscillations that can produce some truly obnoxious squealing. Engineers fight a constant battle to dampen and mute these harmonics by way of shims and gaskets and whatnot.

A certain European make I worked for years ago started using a very soft material in their brake pads which definitely got rid of the noise... but the pads were shot in 8,000 miles.

So they went back to the regular formulation and started shipping tubes of a special sound deadening grease, and that worked really quite well. I still have a few tubes of it sitting around that I use on all cars.

Basically, you can never be quite sure where the various tolerance stack-ups will conspire to allow this noise to develop so a dis-assembly and visual inspection is required. What we used to do was pull the whole brake assembly apart and examine for shiny spots amid the dingy brown. The shiny spots were where the noise was coming from. A little of the magic grease on your finger tip applied to the shiny spots and thin layer on the pad backing got rid of the noise for the life of the pads.

I know it sounds like voodoo, but it works.

What does the Toyota TSB call for?