The carpet HAS to come out in order to do vinyl, the tiles though pose another environmental problem which you do not want to deal with. Usually when you are going to lay another floor down the easiest thing to do is lay down a new subfloor over the tiles and lay the new vinyl down over that. Completely safe and economical. Make sure youscrew the old floor down to the floor joists using the proper length grabbers so that you rid yourself of all of the squeaks. Best of luck.