TomD
Great looking project, and some beautiful wood.

On the squares, some chair makers use that approach to make the front rail flat so that there is less pressure on the inside of the leg. In that case there is a step down form the size of the squares, but the material under the seating is squared (usually rectangular) in section. You can simply carve an airfoil into the rung, but I prefer the method shown here, but with the flatter section, looks neat, and more solid.

Antique:

Boggs: