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Cardboard is common and commonly used (blue Punch boxes, some Macanudo, Don Tomas, some Partagas, etc). It is definitely different from wood but most don't seem to mind as much. it [might] sound a little less bright but overall we use 'em. Build on.
If I were you, I would do this: look on the inside at the side panels. remove some of the white paper if applicable, and see if its cardboard (I call it pressboard. Same thing). Most likely the lid is pressboard. In my opinion, I would avoid boxes that are made of coarse, pressed sawdust like material. Kinda the same material as cheap laminated countertop or shelving is made of. It chips WAY too easy and doesn't saw well.
A wonderful fun book, well worth the money.
As long as the neck goes thru the box it doesnt matter too much what the box is made out of. Although it seems they all have different properties, which adds up to a different sound. I made a guitar all out of cardboard to see if it could be done. The neck is prone to warping though and I ended up only with a one-stringer.
Dunno if this would work and I'm sure it wouldn't work past 3 strings, but if you took strips of .75" cardboard, glued them together to make a 1.5"x.75" neck so that the 'grain' was vertical, I wonder if that would add strength. Would need to be really tight cardboard to begin with, somewhat like a single ply paperboard like a cereal box and NOT this cheap crap cardboard from China.
I could see this as similar to a bunch of tempered hardboard glued together.