Decided to replace the cabin filter today and in doing so fixed the glove box (hopefully permanently). The first pic shows the metal clips I was referring to earlier. They weren't broken after all, but did appear "flat".
I took some needle nose pliers and bent the sides out a bit, in order for it to hopefully grip more once they were in their slots.
When I reassembled the box I pushed it in by hand and then used a rubber mallet to ensure everything was truly secure.
The left side that was sticking out didn't sit well at all after reassembling, but with the rubber mallet, it went back in and appears to be staying.
https://www.flickr.com/gp/99777373@N05/9a9H4p