You can't recycle cardboard if it's soaked in pizza grease. If your local pizza joint puts a piece of wax paper between your pie and the box, you can sometimes get away with recycling the whole thing; if not you can at least rip off the unstained top of the box to throw in the blue bin.
Unfortunately it achieves the exact opposite of what it sets out to accomplish -- it makes the entire pizza box non-recyclable.
The top of the GreenBox rips off and splits into four square shaped "plates". The grease from the pizza ruins the recyclability of the plates, leaving the garbage (or the rare compost pile) as the only option at the end of the pizza party. I'm also pretty sure that four paper plates require less wood pulp and energy to be produced than the four corrugated cardboard squares.
The bottom of the box folds up into a take-away box, which I admit, is kind of neat, but again, it increases the chances of the cardboard being corrupted by grease or juice from the toppings. In the world of good greener design, neat does not trump sustainability.
The GreenBox is a problem wrapped up as a solution in search of a problem.