Is it possible to put vinyl or some such material over small 1 inch square tile that is on an existing bathroom floor?
Yes - You can lay vinyl flooring over an existing tile floor. Be sure that you can live with the level of the existing floor with the additional flooring on it.
Before you start, you may need to apply liquid cement or mortar to fill in any spaces on the tile floor that are broken or un even. You may also need to fill in the grout if the grout is considerably lower that the tile level.
Good luck on your bathroom remodel.