  We show that there always exists an inscribed square in a Jordan curve given
as the union of two graphs of functions of Lipschitz constant less than $1 +
\sqrt{2}$. We are motivated by Tao's result that there exists such a square in
the case of Lipschitz constant less than $1$. In the case of Lipschitz constant
$1$, we show that the Jordan curve inscribes rectangles of every similarity
class. Our approach involves analysing the change in the spectral invariants of
the Jordan Floer homology under perturbations of the Jordan curve.
