Curiously, I completely failed to notice a similar identity to the sumoftwosquares identity, and one more directly applicable to the problem, for the product of two differences of two squares, namely
(a^2  b^2)*(c^2  d^2) = (a*c +/ b*d)^2  (a*d +/ b*c)^2.
As with the sumoftwosquares identity, the product of k factors gives 2^(k1) formally different expressions of the product as a difference of two squares. Alas, the expressions involve terms of total degree k. There is also a "headtotail" property of the various conditions that arise in the problem, which I didn't see how to use efficiently. Taking the example from the solutions
A=[9, 28224, 419904, 3968064]; B=[0, 47952, 259072, 2442960]
and letting M_{ij} = A_{i} + B_{j}, we see that taking the difference of row j and row i of M gives the constant difference A_{j}  A_{i}. (Similarly with differences of two columns)
Taking row 2  row 1, row 3  row 2, and row 4  row 3 we find the conditions
168^2  3^2 = 276^2  219^2 = 536^2  509^2 = 1572^2  1536^2,
648^2  168^2 = 684^2  276^2 = 824^2  536^2 = 1692^2  1572^2, and
1992^2  648^2 = 2004^2  684^2 = 2056^2  824^2 = 2532^2  1692^2
where the "head" of each expression in one set of conditions becomes the "tail" of an expression in the next.
