Not sure how some think that Amnesty for illegal aliens can be stopped without removing Obama from office. He does not need Congress to write an executive order for Amnesty, though it will be unconstitutional.
Obama will just order the Border Patrol to stand down and stop deporting people. The House can not defund people to stop them from NOT doing something. It does not cost any money to NOT do something. It cost less money to NOT defend the border than it does to defend it. How do you defund that?
Congress can write a clause into budget that that prohibits Obama from spending money on the implementation of Amnesty by executive order, but what will stop him from simply ignoring it, if there are no teeth behind it.
Those, who think they can stop a broader Amnesty for millions of illegal aliens without removing Obama from office are just deluding themselves and engaging in wishful thinking. If Obama wants to do it, he will do it. Congress has shown not only that they will not stand up to him, but that the Republican leadership actually wants an Amnesty, also, and they are colluding with Obama to achieve it, without raising the ire of their own constituency.