I'm definitely no expert here, and I'd definitely welcome further input from people who really know what they're talking about, but it occurs to me that mirrors are 'back-silvered'. i.e. they're sheets of normal glass with a silvered layer applied to the back of them.
If you use any form of glue (such as 'No More Nails') you'll actually be sticking the silver layer to the MDF, rather than the glass itself, and I wonder if there's a risk that the glass will pull away from that layer (leaving you with the 'silvering' still stuck to the MDF and a pile of broken glass on the floor).
A more secure way of attaching the mirror to the MDF might be required.