Always remember. Plywood is not an option for humidor building material.
As someone who builds furniture for a living, I beg to differ with your opinion. Even quarter-sawn solid lumber that has been properly kiln dried, (which is the most stable solid wood), still moves, expanding and contracting with changes in external humidity. Depending on the wood used it may move more or less but it will ALWAYS still move. When a craftsman builds something of solid wood there are centuries-old techniques we use to keep this inevitable movement from tearing the piece apart.
Quality hardwood plywood, with moisture-resistant glues used in construction, is vastly more stable than any solid hardwood. Quality hardwood plywood is very suitable, in fact preferable to solid wood, especially for the large flat top, sides and backs for an enclosure exposed to differing levels of internal and external humidity. It can be veneered or covered in solid wood, but plywood provides great dimensional stability under varying humidities both inside and outside the humidor.
Without seeing this box I cannot be sure, but it appears to me that it is veneered MDF.
Medium
Density
Fiberboard is essentially, sawdust mixed with glue under compression to made a very cheap substitute for real wood plywood or solid wood. This material is very inexpensive but the MDF expands radically and turns into sawdust when exposed to water and the glue holding the sawdust particles dissolves and breaks-down and the wood particles expand. I believe that somehow the bottom of the box became wet on either the inside or outside. (From the looks of the damage it seems likely that water soaked-in from the outside. Perhaps rain running across the surface it sat on, a spilled drink, condensation from the outside of a glass filled with ice... anything like that could cause this type of damage in MDF.)
Hope this adds some information to help solve the puzzle and some things to keep in mind as you contemplate a possible wooden solution to your cigar storage problems.