I think the answer is YES .
Suppose that $d$ is a little square that lies in a unique maximal subrectangle $C$ in the polygon. Then any anti-rectangle contains a unique little square in $C$. We can replace this square by $d$ and still have an anti-rectangle.
So one is required to prove the following:
Proposition: Some corner square must lie in a unique maximal subrectangle.
Sketch of proposed proof: Observe that as one moves clockwise around the perimeter of the polygon, one must go either left or right at various stages. One obtains a sequence: $R,R,L, R, L \cdots$ and it is clear that the number of $R$'s is 4 more than the number of $L$'s.
Motivating observation: suppose that in this sequence one has three $R$'s in a row. Then the corner corresponding to the middle $R$ will lie in a unique maximal subrectangle.
The problem is that one may not have three $R$'s in a row - one does, however, have at least $2$ $R$'s in a row. Let us examine whether or not one of the two corresponding corners lies in a unique maximal subrectangle.
(Here my sketch will get icky, because I don't have Joseph O'Rourke's excellence in drawing pictures...)
I claim that the only way both of these $R$-corners can fail to lie in a maximal sub-rectangle, is if the piece of the polygon 'opposite' the corners has some kind of crenellation, i.e. two knobs sticking out opposite each corner. These might not be 'smooth' - there could be many many bends in them but, still, if one gives this a little thought it becomes quite clear that this can only happen if `opposite' the edge between the two consecutive $R$-corners one has two consecutive $L$-corners.
Now we know that this cannot happen to all consecutive $R$-corners, because the number of $R$-corners exceeds the number of $L$-corners by $4$.
QED