There is an affirmative answer for rectangles (and therefore also for rectilinear polygons) in the paper "Rectilinear Glass-Cut Dissections of Rectangles to Squares", by Czyzowicz, Kranakis and Urrutia.
Note the different uses if the word “rectilinear”: Wikipedia defines a rectilinear polygon as one with rectilinear sides. The paper above uses rectilinear to mean that the rectangles can be produced using only rectilinear cuts, without needing to make any right-angled turns in the middle of a cut.