Kudos to you for planning ahead and saving compost materials!

You certainly could do something like that. You might wish to punch a few big holes in the sides to encourage air exchange. Compost needs to breathe and I don't know that a closed box (especially if it's damp) will allow enough air.

A decade or so back I remember a thing about composting in paper grocery bags. I don't know what the result was exactly--I think it was pretty mediocre so the whole thing pretty much faded away.

I think the main argument against composting in a cardboard box is that compost piles tend to work better if they're BIG--minimum 3 feet in every dimension. Even if you stack boxes next to each other it isn't the same--materials won't mix easily, and the cardboard will form a barrier.

Do you have a shady spot where not much grows (you won't want to put a garden there anyhow), or a stump that needs to get rotted, or a weedy spot that you don't like? Those would be good places to start a pile.

I would certainly tend towards just setting up a simple wire fencing compost bin in one of those types of spots and starting your pile there.

You will be unlikely to achieve the optimum levels of heat from that amount of space. You might be better served by a vermicomposting system until you figure out your garden layout. Typically the most efficient design is to have the compost pile near the garden so you don't have to spend a lot of time and labor adding and removing material, and convenient to the house so as much as possible gets added to the pile.

The card board would break apart and not really hold the compost for very long. If you don't want to have a bin, you could just make a big pile.

Since many compost piles are very portable it really matters not where you put it as long as it is someplace easy for you to get to. I have had compost piles in several places as I found the place that was most convienient for me. The concrete block one built at the back of the garden was too far away in the winter when wading through 4 foot snow at minus 10 was not what I really wanted to do, although is was really handy in when I spread the compost. Pretty much the same with the wire bins in the middle of the garden. The bins now along side the yard tool shed are easier to get to, even in the winter, even though that means trucking compost further when the plnating beds need some.
Every location of a compost pile is a compromise, and you have an opportunity to try several to see what works best for you. That bompost pile should be convienient for loading, even during adverse weather, and convienient for unloading.

I've done it several times to tuck compost behind other plants. I just opened up the bottom of the box to splay outward & piled materials inside. The upright sides didn't break down as quickly as one might think.

I don't think you'd be able to move the boxes of compost materials around once wet & mixed. You could use your boxes for storing shredded paper dry until you create a large pile outside somewhere.

Another use for boxes is layering over turf then add compost in the boxes on top of the sheet of cardboard. In 6 months or so you'll be able to plant in the boxes. We used new boxes as raised beds for potatoes & it helped keep the mulch in place as we hilled them.