What does "Request Boxing Vectors" means?
Boxing vectors is not a standard term. I can't tell you exactly what was going through the mind of the pilot (you would have to ask them directly), but based on your description, my interpretation would be that the pilot was requesting delaying vectors to give them time to work checklists. It is pretty common in an emergency situation where the pilots need time to evaluate the situation and plan their actions to request either a holding pattern or vectors around the airport. Boxing vectors could refer to the pilot wanting to be put "in a box" of airspace and left there until they had had time to figure out the gear issue. That would be my guess if a pilot requested boxing vectors on the frequency (although I would ask them for clarification of course).

In an emergency situation, the pilots don't really care where they are going (as long as they stay relatively close to the airport) so it makes sense to let it be up to ATC to decide. ATC then has the opportunity to move the flight to a quiet piece of airspace where they are not in the way for other traffic and let them run the checklists. It's a way for the pilots to reduce mental workload, as all they have to do it punch in a new heading every few minutes, instead of calculating and setting up an holding pattern in the navigation computer.