Our condo has old kitchen tiles that may be asbestos how to deal with this?

You should have the broken tile and glue tested in a certified lab (just look in Yellow Pages). If not asbestos, you can do what you want. If it is, you really need to have it professionally removed. Have at least have three companies give you estimates. You can do the plywood deal, but the tiles will continue to break as they age and the linoleum will not be much protection. You would need to seal the plywood to the wall around the edges will some sort of caulking. Even with that, you would need to have it removed if you ever sold or pay the buyer to have it done.