I am reading the article On Goursat's Proof of Cauchy's Integral Theorem written by Harald Hanche-Olsen. In a historical remark, the author wrote that
Goursat presented a proof in 1884 for the case of a simple closed curve, by dividing up the interior into small squares...
Moore cleaned up the proof a bit, paying more careful attention to the treatment of the boundary curve. He also introduced the current idea of proof by contradiction, by subdividing and always selecting a part where the desired conclusion is maximally violated, and then applying the definition of the derivative at the resulting limit point.
Pringsheim presented a more severe criticism of Goursat's treatment of the boundary curve in 1901. He pointed out that these problems disapper if the proof technique is applied to a simple geometric figure such as a triangle.
Since some modern textbooks (such as Bak and Newman, Cartan, or Beardon) do use rectangles to present Goursat's proof, I don't understand what is the issue with rectangles. What is Pringsheim's criticism all about? Is his criticism justified? Is it OK to use rectangles?
I think you're misreading this. As I interpret it, what he says is that the proof technique of subdividing into rectangles works fine if the boundary curve is a triangle, but that there is an additional argument needed to get from there to a more general boundary curve: “Then the theorem follows for simple polygonal paths in a simply connected domain by triangulating the interior of the path, and finally one gets it for a general path by approximating it by polygonal paths.”