Here is another photo

You have a couple of problems here...possibly. My best guess is that the mesh-like pattern you see beneath the tile is either 1/4 masonite underlayment or 1/8" tempered masonite underlayment. If the tile is hard and brittle and 9" X 9" 1/8" guage, then it probably contains asbestos fibers. If is is flexible and only 1/16" or less guage, then it could be Goodyear vinyl tile. I have seen many Goodyear vinyl tile jobs improperly glued down with the black asphaltic adhesives over that type of underlayment panel. Since there appears to have been almost no transfer to the back of the tile, my guess is the black adhesive was the water based emulsion-type, which contained no asbestos. If of the black 'cutback-type' adhesive, it 'might' contain the fiber, but you would have to do an awful lot of work to get a fiber out of the glue matrix and into the air. An industrial laser might do it.

The HD contractor is being cautious about not touching the stuff for liability reasons (running afoul of federal EPA rules and criminal prosecution). Rules concerning removal of asbestos containing (AC) floor tiles have changed, however. I believe it now again legal to manufacture and sell AC floor tiles, although the flooring industry will probably stay away from the raw material. Do an internet search for rule changes.

The tiles would pop off very easily, although as a professional I cannot recommend you do that. Many years ago we would rip all of it out...tiles and underlayment...and start fresh. It was not a big deal.

If covering it all over is no big deal for you, then probably that is what you should do.