Did the guy at Home Depot tell you the existing tiles might be asbestos? What gave him that idea? Did you bring a sample with you to the store?

Some things that may narrow it down..... When was the sun room constructed? If the sun room was constructed in the 1970's or earlier, the tiles may contain asbestos. If the sun room was constructed sometime in the 1980's or later, chances are there is no harmful asbestos. Do you know the age of the existing tiles?

If the existing tiles are still in good shape and firmly attached to the concrete slab, you can still install your new slate tiles right over them without any real worry. It's not going to cause any future problems when you sell the property. Besides, no home inspector for a buyer is ever going to remove any of the slate tiles to check what is underneath.

Asbestos exists in many homes without the homeowners even knowing. The joint compound used to finish all the walls in homes during the 60's and 70's had asbestos in it. I don't think there is any legal issue in covering over the existing tiles.

You never really mentioned the condition of the existing tiles. Are they still firmly attached to the slab with no problems? Are any coming loose? The condition of the existing tiles is important because if there are problems with their adhesion, that will need to be addressed prior to any new floor being put down. Specifically a possible moisture issue on the concrete slab.

I wouldn't really start worrying too much at this point. Provide a little more info on the current condition of the existing tiles. That would help a little regarding your next move on the situation.