Unfortunately, stishivite isn't stable at STP. It quickly degrades into alpha-quartz. Alpha-quartz might still make a decent building material, but it doesn't have the density advantage.
You should search for what they did to hide the excavated material (called "spoil") from NORAD. Most of it was shipped out by truck at night and used as fill for other projects. It's also rumored that they used underground rail spurs to transport it.
If the digging organization is well financed, I recommend trucking it to somewhere else and pretending that you're mining the material there.
Addendum I did a little digging based on Kevin's commentary. Based on This article, stishivite will stay stable for thousands of years at room temperature, as long as you don't get it wet. If you add water, it'll decompose in a period of days.