I have been trying to keep up with it by shredding and/or layering it in the garden.

I got an idea, not sure what I need to do exactly. I have an old cattle hay ring, about 12' in diameter. I'm thinking about just breaking the boxes down so that they will lay flat and tossing them in there to make compost. The ring will keep them from blowing away.

I won't have any greens to add over the winter to help break it down. Any suggestions?

Failing that, you need some greens unless you're just going to let it go slow. Can you put a small family of pigs in there with it? :-p

I don't mind it going slow. As long as I don't run out of room! Could I add something like urine? I don't mind peeing in a bucket or adding while I'm in the yard. I have 10 acres so neighbors aren't going to see me!

Maybe throw a few shovel fulls of dirt occasionally? I have plenty of fall leaves that I gather with my leaf vacuum, but they are browns too and I want to put them directly on the garden.

Soil is not really going to help, if it is already in contact with the ground it will have all the microbes it needs. Too much soil will actually slow it down by reducing air flow to the compostables.

Leaves (as you probably know) decompose slowly by themselves to make a great leaf mold compost. I would think that a cardboard/leaf pile, left for a full year at least, would make some great compost loaded with worms.

I use cardboard in various ways. When making new beds in the lasagne style, or simply laying it down and covering with hay or grass clipping to expand a bed. I lay down cardboard and cover with an inch or two of wood chips for pathways. I use large pieces while painting or staining to set up materials, and to lay down along edges, in lieu of a drop cloth. I also lay cardboard down to line up my winter-sowing containers, instead of them sitting on the dirt.

Worms and other critters love cardboard! I found a salamander under the cardboard and winter-sowing containers once. (Slugs probably hide under there too, not so great.)

If the boxes are reusable, you can put them up on Freecycle or Craigslist Free, and people who are moving will happily come pick them up. This may depend on how rural you are though.