If you want it to keep out predators, cardboard is a very bad idea. Any predator can get through cardboard.
It will get soggy and fall apart fairly quickly in rain.
But, as long as it stays dry, cardboard does work just fine to block wind and provide shade from the sun. And it will keep off a few raindrops here and there.
So whether it is a bad or a good idea will really depend on what you want it to do. If you have a run that is completely predator proof, putting cardboard inside to provide some shade and wind protection could be a good temporary idea.
Given how fast cardboard comes apart when it gets wet or even damp, I would not try to make any kind of a real coop from it. Just take a big box, turn it on its side, and put a few bricks or stones in the bottom to keep it from blowing around.
I agree that is cute. But if you are going to take the time to build something that elaborate, I think you should do it with sturdier materials, not with cardboard.