re: Building the Boxes

Tape on the inside works well for the squeeze out but well noted on the problems removing it if it gets caught into a corner.
As an alternative I like a smear of paste wax. Wipe on generously by swiping out toward the edge to avoid the cut edge from scraping excess wax off of the rag. A small amount might bleed onto the cut but that will not hurt. A hard wax like paraffin will also work just more effort needed to remove all traces given the difficulty gaining access to tight spaces.
Best to avoid wax with open grain woods like oak as the wax can be tough to fully remove from the pores.

Removal with mineral spirits will allow a finish to stick, but typically I'll only use the wax on boxes I don't intend to have a finish on the inside (lingering odor and difficulty applying a smooth coat). Also makes the wax cleanup less critical.

Typically the glue will peel off in nice strings.
You can also use tape, leaving the smallest gap you can between the tape and cut, then apply wax as described.