Guacamole[0] supports SSH connections[1]. This isn't running SSH over HTTP, but does result in a terminal in a web browser. In my experience while simply changing the listening port does work in general, when a network operators is using DPI or similar, they have typically failed closed. If they're unable to intercept the traffic it's blocked.
[0] https://guacamole.apache.org/ [1] https://guacamole.apache.org/doc/gug/configuring-guacamole.html#ssh