Pretty hard to process paperwork with an empty office.
If Obama wants to play hardball, the house can simply furlough all immigration services employees that process immigration paperwork.
How do you do that. You simply defund the employees that handle that paperwork.
Nobody gets a green card. Nobody is processed. End of argument.
Obama comes up empty.
No H1B visas either: good way to peel off some of the Chamber of Treason types from a general amnesty push.