What's wrong with using rectangles rather than triangles in Goursat's proof?
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.”