How do I cover the mirrors on the back of the china cabinet?

You could just use the textured wallpaper that looks like the wainscotting panels. It can be painted any color. I used the new removable Contac paper on the back of my china hutch. Worked great.

I would use contact paper or peel and stick wallpaper with a cool design over the mirror. It could really liven up the cabinet and would be easier than cutting and installing wood paneling.

Another option would be breadboard, depending on the look you are going for. I assume any strong adhesive would be enough to hold it to the mirror. You could also install a small trim piece around the edges for extra security, but as long as what you install doesn't have much weight, some kind of gorilla glue should work.

Those mirrors are usually glued onto the cabinet back over a thin wood. If its the mirrors you don't like, you will have to remove the shelves, tape it, and break it up and remove. Clean off glue and paint the back, or use some cool paper on it.

Then if your hoping to just have the wainscoting showing through, use a drill to make two hole big enough to put in a jig saw blade through along where the top of your wainscotting will be. I will say this will make the cabinet less stable, so on the back out of view, I would add a piece of wood like a 1"x 1/2" piece of lumber screwed into the sides, and any center piece to add stablility.

Liquid nails will hold it.

Cut the wainscoting to fit and use command strips to hold it in place. When you are tired of it or want to change it, it will be easy-peasy. Another thing you could do is cut cardboard to fit and cover it with wallpaper, cloth or whatever, and put it on the mirrors with the command strips.