I stand corrected on the chemical makeup of Titebond glue. However, I've seen too many soak-tests to prove that while III may be the best water barrier, II is superior for immersion applications.
Not that this is important for beehives unless you intend to drown them. :)
Regardless, glue used in beehives should ideally be of the low-VOC type. Construction adhesives are inexpensive and easy to apply from a caulk dispensing "gun." They can be thinned or thickened, as needed, and spread over some sort of woven material, such as fiberglass or drywall tape to repair broken edges, enhance strength at corners and/or provide water resistance. At US$3-5, brand dependent, per tube able to be used on many boxes, it's a good investment. This is particularly true if you have lots of exposed end-grain (butt-joints) or lots of seepage surface (finger joints) at your box corners. It has the added advantage of reducing your reliance on metal fasteners, if you are one who cares about such things.