A hostage crisis?
Well, it is certainly heated rhetoric, but it's not far off the mark. In a setting of good faith negotiations, both sides have to make concessions, ie. give up some of the things that they want, to reach a compromise. Republicans aren't offering any actual concessions though. They're saying "give us what we want or we'll blow up the economy". Make no mistake, while the shutdown is bad enough, what would really wreck the economy is not raising the debt ceiling. The GOP caucus, excluding a few of their craziest members, don't actually want to push the US government into debt default, so they're not making any concession at all if they agree to raise the debt ceiling. They're just playing a high stakes game of chicken.
Agreeing to Republican demands with regards to the Affordable Care Act would set a terrible precedent. It would mean that any time the nation approaches the debt ceiling, any and every law already on the books is up for grabs. The laws wouldn't even have to be confined to budget issues. It could just as easily be "do what we want on this foreign policy issue" or "change this environmental regulation law" or anything else in exchange for not ruining the economy.
As for the ACA, it's already a compromise law that includes, among other things, over 100 Republican amendments and no public option (which most Democrats wanted in there). It passed the House and the Senate, was signed into law by the President, and was upheld by the Supreme Court. To now allow the House to in effect unilaterally change it -- so they won't blow up the economy -- would be a terrible abuse of the government's system of checks and balances. Why have the Senate and the President involved in lawmaking at all if the House can later simply extort both to get whatever changes they want made to existing laws?
We have a hostage situation.