Why are the Jesuits and Society of the Sacred Heart gender specific? Why is it impossible for women to join the Jesuit order but must instead join Society of the Sacred Heart?  Is there something in the initiation ritual that would exclude women?
The Society of Jesus and the Society of the Sacred Heart are both religious orders. All Catholic religious orders are by definition either for men or for women.

As to why this is... Well, historical Catholic understanding gave very different roles to men and women both in normal life and in the sphere of religion. Men, for instance, had priestly roles, preaching roles, roles teaching in universities and the like. Women, by contrast, had roles that mainly consisted of charitable works and, in recent centuries, the education of girls. They were distinctly different roles, and therefore had entirely different orders.