Once dry and stable, clean up all the excess glue with a razor blade. Wrap the cushions in newsprint and tape, and give the whole thing a thorough sanding. The main purpose of this last finishing is to get all the dowels perfect flush with the face of the plywood. You can fill little gaps with Liquid Nails and a putty knife, let cure, and sand down. I then put on three more coats of hand-rubbed polyurethane, including working the rag down into the gap between the pieces and hitting all the dowels. Once dry, sand with 330 grit or higher, then wipe clean with mineral spirits and hand-buff with a lint-free cloth and some furniture wax.
Sit down and take a break.
I wrote this Instructable while sitting in the Four Square Chair. I would recommend it for an office or dining-type situation. It is quite comfortable, but it is also rather upright for relaxing. Should be pretty indestructible -- aluminum and plywood are incredibly strong.
If you want to save yourself the trouble, the one and only Four Square Chair currently in existence is for sale here: http://www.etsy.com/view_listing.php?listing_id=10798264