Until the new Senate is seated whatever he could do through the House would be pointless. The issue is the Executive Branch usurping the role of the Legislative Branch which has to end up before the USSC at some point anyway or else it's just kicking the can down the road.
What Boehner does when the next Congress is sworn in and in session will tell the tale, not what he does at the moment. I don't care for Boehner but getting the current democrat Senate on record as avoiding the issue won't accomplish anything at this point.
Any unlawful EO can send the States directly to the Supreme Court for an injunction. That is something congress can't do.
The States have a role in this.
Until there is an unlawful EO, the States can't do anything like that.
/johnny