The issue is not the substance of the Republicans’ argument, it’s the tactic. There’s an established procedure for putting in place a law – it passes both Houses of Congress, and is signed off on by the President. Once a law has been passed, it can be repealed or amended through the same procedure. Everybody playing by that set of rules helps keep arguments fair and procedures somewhat predictable.
What the Republicans are doing now is to say – we can’t win both Houses of Congress and the Presidency (which is what it would take to overturn a law that already was passed by all three institutions), but what we can do is use control of the one house that we do have to refuse to fund the government until the President and the Senate change the health care law.
The reason that the President and the Democratic Senate are refusing to negotiate (and should continue to refuse to negotiate) is that they don’t want the House of Representatives to get in the habit of shutting down the government everytime there is some law they disagree with. It’s essentially the exact same argument for why it’s generally considered inadvisable to negotiate with terrorists. There’s a legitimate way for a party in Congress to get its way – through the normal procedure of writing bills, and through the election cycle – and there is an illegitimate way to do so – by holding the government’s operating budget hostage.