Great guesses. This piece is very likely part of the copper-rich quartz vein they were hunting at the champion mine. So I would say its not quartzite, but quartz since it deposited in a vein rather than being metamorphosed into a quartzite.

As for the greenish blue mineral, the locals called it azurite. I doubt that, it's the wrong color. My guess is chrysocolla, which is a hydrous copper silicate. There were no sulfide minerals spotted in the area.