Basement flooring. Tile?
Sounds like you're searching for someone to tell you what you wanna hear? You need to remove those god-awful tiles and start from scratch. Use a tool similar to an ice chopper, or you can rent an electric tile remover machine, made to remove vinyl, asphalt tiles even sheet vinyl and rubber backed carpet.
Once all the tiles are removed you can install tile with certain products and procedures. Try not to make dust from crumbling tiles, (most likely they contain asbestos if made before the mid-80's), simply removing tiles should present no hazards at all. Just do not sand the tiles or adhesive below, especially if the adhesive is black.
Installing ceramic tiles over certain vinyl tiles is possible with certain products, however the entire process is dependent on the bond of the old floor. Kinda risky, and all the thinset manufacturers want you to prepare the old surface, meaning sanding....which you do not want to do.
Tell me about this "contractor" that suggested Double Duty mastic for this installation...certainly he isn't a tile contractor or anyone that know much about installing tiles!