How can I keep leftover drywall mud from drying out in the box?
I don't have any experience with boxes, but have quite a bit with the 5-gallon buckets of compound.  Without knowing much about how those boxes seal, I'd recommend going with a bucket- the buckets/lids seal air tight which is important for keeping the mud from drying out.  The other thing I'd recommend- every time you close the box or bucket, put a piece of plastic sheeting over the top of the mud.  This traps the moisture in against the mud, and prolongs its shelf life.  The buckets come with a round piece of plastic sheeting on top of the mud- 
I just fold it in half (to keep any bits of mud stuck to the plastic from being exposed to air) then when I am done for the day, unfold it, and put it back on top of the mud in the bucket.