If no one else is going to go after this one... I can give an answer based on, well, working in the spirit of the best solution the program could find. But I can't prove that there aren't any larger squares that can fit in the unit cube.

plasmid's post in Calculated was marked as the answer
July 20, 2013

Now we just need seven more of those to make the whole tongue-twister.

Without any sort of formal proof, I think that these are all possible shapes. Although I haven't yet fully convinced myself that the last shape I drew actually exists. It seems like it should though.