Using the convention of the wikipedia article, Cauchy-Binet formula states that --for $A, \, n\times m$ and $B, \, m\times n$ matrices--
$$ \det(AB) = \sum_{S\in\tbinom{[n]}m} \det(A_{[m],S})\det(B_{S,[m]}) $$
My question is, is there a formula for the "squares" i.e.
$$ \sum_{S\in\tbinom{[n]}m} \det(A_{[m],S})^2 \det(B_{S,[m]})^2 = ? $$
Many thanks