They can't eat it while it is dry and intact, but in the conditions you are describing it would not stay dry and intact for very long. If the cardboard box is kept outside it will get rained on. And even indoors, a cardboard worm bin would be constantly in contact with worm bedding at 70% moisture - i.e. as moist as a "squeezed out sponge" - even a squeezed out sponge would wick a lot of moisture onto dry cardboard. The cardboard would get soggy and collapse before very long.

If you wanted to make a simple outdoor worm bin (which may be what you are getting at) rather than a compost pile on the ground, you could use some kind of wooden crate, chest or box, or build one of wood pallets or other scrap wood. Wood often works well for an outdoor bin because it is a little bit permeable to moisture and would allow excess moisture from the bin to evaporate when the temperature or humidity fluctuates. Wood would eventually break down too but much more slowly than cardboard would, and a wood bin can also be repaired or patched if some of the boards deteriorate, extending its life.

A wooden worm bin can use a tray system, or be even simpler like a chest or box. Here are some images to give you an idea.

https://www.google.com/search?q=wooden+worm+bin&tbm=isch&tbo=u&source=univ&sa=X&ved=0ahUKEwim4JilyIzVAhUIHD4KHYkgCSYQsAQISw&biw=1440&bih=803

I would consider a cardboard box in my dry zone where it does not rain in the summer. When it collapses, it could be replaced with another box.

Kinda like me living in a ginger bread house... yummm

If the cardboard box is in contact with the soil it will absorb moisture regardless of how dry it is in your area. Even in a desert you can get water from the earth... simply dig a shallow hole, put a container in the middle of the hole and then place a plastic film over it with a small pebble in the center. Moisture will be drawn from underground to the surface where it will evaporate and then condense onto the inside surface of the film where water droplets form which then will run down the film to a low point created by the weight of the pebble at which point the liquid water gathers into a heavy enough drop to drip off the film falling into the container below it. Try it, it actually works. Its the reason why soluble minerals such as calcium and sodium may be found lacking in areas of high rainfall yet are always in excess in arid regions... Where rainfall is low water moves to the surface bringing certain cations with it (quite the opposite of what happens to those same cations in areas of high rainfall that subsequently get washed into the subsoil layer). Also even without rain dew can still form if temperature conditions are right for the humidity... this is what keeps some cacti going in rainless years.

Now for the more practical answer... how you gonna keep the worms alive without a moist bedding? You essentially are making a bin out of bedding and wondering how long it will last... my response is "not long"... now back to that ginger bread house, we could use sugar for the window glazing and gum drops for the door knobs... and chocolate shavings for roof shingles... with all this talk about food, I gotta stop now and go eat some cheese cake.