This is nonsense. If the ACA included such a provision, it was only because Congress wrote that language, and the U.S. House can remove such language.
So they can. But it won't become law until passed by the Senate and signed by Zero.
The article does raise legitimate questions.
The House can, by itself, refuse to authorize expenditures which haven't already been passed into law. That's the effect failure to pass a CR would have — what's already enacted proceeds; what isn't stops. Unless it's found to be "necessary" (e.g., air traffic control) ... that allows plenty of scope for executive-branch shenanigans, as the article notes.
The House cannot, by itself, repeal laws that are already in effect. If the CR, as originally passed by the House with Senator Cruz's recommended rider, were approved by the Senate and signed by Zero, then ObamaCare would be out (of course, why not just repeal it?). But that's not going to happen. So, the question is, where from here? What is the actual effect of not having a CR?