Note that this is not squaric acid as it has no double bond between the carbons holding the hydroxy groups.
I know this isn't exactly an answer, but I think it might help in reaching a qualitative understanding. These are the geometry optimized structures of the protonated and singly deprotonated form of the molecule (I could not obtain a theoretical pKa; it is a much more involved calculation than I anticipated). I would expect the pKa to be fairly low, as the deprotonated structure would seem to be fairly stable due to hydrogen bonding.
(Images were obtained using GaussView. Optimized structures were from Gaussian Optimization calculation using APFD, Basis set 6-311G+(2d,p)).