Many installers use a construction adhesive to mount mirrors to the wall. Although there is no mounting hardware to show, it makes it very difficult to remove the mirror in the future. You'd have to force a putty knife between the mirror and the wall, being ready to catch the mirror when it pops loose. But be aware that there is a good chance that the adhesive will pull off pieces of wallpaper, or the paper surface of the drywall. Unless the new mirror will entirely cover the area of any damage you cause, you'll have to make some repairs.
If you are still determined to replace your square mirror, you could instead try using a length of music wire with each end wrapped around a wooden handle of some sort. Put the music wire behind the mirror and pull the wire back and forth, so that it will cut the adhesive like a saw. Eventually you should be able to work the wire all the way across the back side of the mirror to break it loose.
When you put up your new mirror, you might consider using plastic mounting clips with screws that are made for this purpose.