Greg, if the tile is from then, there is a good chance it does. You can buy home testing kits at places like Lowes or Home Depot. They are not really that expensive and better safe than sorry.
Perhaps they may. If you go on line you can purchase low cost testing kits for asbestos. You simply break off a few inch piece of the tile, put into bag and for around $35 they will tell you what you have.
Asbestos floor tiles are on the low end of risk. Making them easy to remove for the company that does this type of work. Although some containment is required when removal is going on, Its not nearly as difficult of a job when pipe insulation that contains asbestos must be removed.
Removal of this is not nearly as costly as you would think. In any case, get the tile tested and then you can make the necessary decision on how to proceed.