You definitely want to use mirror adhesive that is approved by the maker of the mirror tiles. The reason is, some adhesives will attack the backing on the mirrors and end up showing through the mirror. Follow the adhesive instructions carefully and it will work as designed. I have seen mirror tile installed with general purpose liquid nails with a blob in each corner and the install looked fine, for about a three days or so. Then four brownish blobs started to appear on each tile, one in each corner. The blobs eventually turned greyish-brown and the mirror over the blobs was gone, it was just semi-transparent glass at that point. The homeowner who did it was quite distressed but had covered an entire wall of her living room with them and decided to live with it instead of getting into the problem of taking them off the wall and what damage was going to be to the plaster wall.
Any of those products you mention will work fine a substrate. They each have different pros and cons though. Is this mirror you are making going to be in a damp or humid location? If so, MDF and especially particle board are prone to absorbing moisture and will swell up and eventually begin to crumble. Plywood can handle humidity better but it can still delaminate. There are plywood products made for damp and wet locations, such marine plywood. It is pricy.
How are you going to mount it? MDF is the heaviest of the three and depending of the thickness can be substantially heavier. Plywood will tend to warp more, especially those with fewer laminations and lower grades of plys like common CDX (a "C" side, a "D" side and e"X"terior rated).
OSB (orientated strand board), which you didn't mention, will warp less than plywood because of its smaller "strands" of wood. It will work better than MDF and partial board in damp locations but worse than plywood. Tends to be cheaper than plywood and is used a lot in the US now since it is made from fast growing trees and a lot of adhesive.
Plywood, and OSB hold screws better than MDF and particle board. All of these products will expand with enough moisture. They are all dimensionally stable if they stay dry. Particle board tends to off-gas a lot of formaldehyde. Not so good for the IAQ (indoor air quality) and could interfere with some adhesives. OSB, Plywood and MDF all off-gas as well, plywood the least and can be obtained with non-formaldehyde adhesives.
Another option is MDF with a veneer but the veneer is so thin and other than the cosmetic appeal of the veneer, it may as well be MDF since it has all the same characteristics.
If it where me doing it, I would choose a cabinet grade plywood with a lot of plys to resist warping. Since it will be covered with mirror, the plywood finish isn't that important and imported baltic birch plywood can often be found for reasonably good prices. I would also make a frame of birch or poplar and biscuit and glue it to the plywood edges to hide the plys. Or, if it was on the cheap, getting edge banding with heat sensitive glue already on it and ironing it onto the edges will hide the plys just as well and can be done quickly and onsite since it hardly requires any tools to do. You can get edge banding at Rockler or Woodcraft in the US. I would seal all of it prior to installing the mirrors to reduce the absorption of moisture.