You can use paper bedding, but I don't think your box will last long!
It will get soggy and smelly pretty quickly.
I use a plastic box with a dropped side to make it easy for the piggies to hop in and out.
I line the bottom with newspaper and then put large piles of hay on top - they spend ages in there munching and nesting, and I would day it catches around 80% of their wee.
A plastic bag is a bad idea, the piggies would eat it... you either need a plastic tray or to replace your cardboard box once or probably twice a week! We have Rosewood corner rabbit litter trays that we fill with hay and the piggies wee and poop mainly in there (I am lying, they wee and poop everywhere because they are guinea pigs, but more goes in the trays than on the fleece!). Sometimes they get cardboard boxes full of hay too and they have a hay room lined with newspaper then paper bedding then lots of floor hay. If they get a cardboard box its so soggy the bottom dissolves and falls of in 2-3 days...