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.

If the glue is old the tiles will probably break up when you start nailing. I had to take 5 asbestos tiles at a time from my dad when I was a kid and throw them in a truck down below being sure not to cross the chalk line dad drew. A number of yrs ago dad & I cleared tiles out of church. Dad died in his 90's of heart problem brought on by high BP. I have never had cancer, I would not want those tiles under the wood floor. A wide putty scraper should pop them loose, use dust mask & gloves and have your kids at friends for day & your problem would be solved. You are not going to saw, it grind it or sand it so fibers shouldn't be a problem except to guy who wants to charge a fortune to remove it! I helped my dad with lot of other removal of tiles so that is just 2 examples, dad did hard flooring for at least 30 yrs. A flat shovel might work well too. I did my wood shingle roof that way.

It is corrugated asbestos on the ducts, and asbestos paper on seams and the register boots. Does the state in which you live have a certification / license for asbestos consulting? If so, please find a good consultant. Many other things in you house may contain asbestos (e.g. drywall joint compound, sheet vinyl flooring, ceiling texture, stucco, window putty and more).

Asbestos is definitely not good for you. BUT: The people who got sick and died from asbestos were the ones who worked with it every day for their entire working lives. It was mostly those working on the factory floors of companies that made asbestos products, and to a lesser degree, those who installed asbestos products day after day. Homeowners have nothing to fear from asbestos, but they've been taught to fear it by TV and social media that thrive on "engagement" -- getting people to sit up and pay attention. The quickest route to that is to make them afraid. QED. You've probably had more negative health effects from the stress you've experienced because of asbestos worry, than from any slight exposure you might have had to asbestos itself. The fact is that one exposure to asbestos -- if indeed you were actually exposed -- is not going to harm you in any significant way. So sleep easy. If you want to do something concrete now, you can launder the clothes that were in the laundry room, and wipe horizontal surfaces with a damp rag. But really, you don't have anything to worry about. PS -- While I don't recommend demolishing possible asbestos-containing materials carelessly, one such event in your lifetime isn't going to hurt you. Besides, you can remove asbestos tile with no asbestos exposure at all just by making sure you keep the tiles wet while you're working with them. If your floor work was done by pros, most likely that's what they did. PPS -- Not all old floor tiles contained asbestos. Only a test can confirm its presence. Asbestos was found more often in the high-end premium products, which were advertised as fireproof. (Old houses had and often still have lots of asbestos in them, with no harm to their owners. In fact that's one reason that when they catch fire, old houses typically burn slower and cooler than new ones.) Asbestos was commonly used in the mastic adhesive (cutback) used to glue floor tiles down. But again, there are ways to remove it without creating any asbestos exposure at all.