I knew that I wanted to construct my four joints in such a way that they could be combined into a square. I also knew that I wanted two of my joints to be strong, but not fully secured, so I could separate the square into two halves as well.
I determined the 4 joints I wanted to create were: Finger Joint, Lap Joint, Miter Joint, and another Miter Joint reinforced with a Biscuit. To create these, I first cut the finger joints by removing material with the table saw. The lap joints were created in this same fashion, and once I had completed those, I took the 4 wood segments over to the Miter saw to give each piece a 45 degree cut on one side.
My plan was to glue the miter joints together to form the two halves of my square. The finger joint was a tight enough fit that it would hold together via friction alone, but the lap joint had no way to be secured in a temporary fashion. My solution to this problem was to use a dowel. I drilled matching holes in the overlapping ends of the lap joint, and press-fit a dowel into the lower piece, so the upwards piece has to be lifted up over the dowel and pressed down onto it for a snug fit.
Because the motion of locking the finger joint into place is exclusively horizontal, and the motion for locking the lap joint is vertical, when connecting the whole square together, you have to twist the pieces a bit to get the two joints to align at once. This twisting motion relies on the flexibility of the wood, but is also what ensures that the square remains sturdy when locked together, because the pieces needed to be torqued to be split apart.
I am quite happy with how this design turned out. I accomplished my goal of making a sturdy square that can easily be split into two and joined back together without any tools required.