Boehner wanted that strategy because he thought Republicans had more leverage on the debt limit than they do on the shutdown. A shutdown, after all, is just bad for the economy. A default is catastrophic for it. You'd have to be insanely reckless to permit the federal government to default on its debts. And Boehner believes that House Republicans are insanely reckless and that President Obama isn't.
But that strategy failed. Boehner's members refused to wait for the debt ceiling. They want their showdown now. And that's all for the better.
Moving the one-year delay of Obamacare to the CR maximizes the chances of a shutdown but makes a default at least somewhat less likely. If a shutdown begins Monday night, Republicans and Democrats will have more than two weeks to resolve it before hitting the debt ceiling.
As Alec Phillips put it in a research note for Goldman Sachs, "If a shutdown is avoided, it is likely to be because congressional Republicans have opted to wait and push for policy concessions on the debt limit instead. By contrast, if a shutdown occurs, we would be surprised if congressional Republicans would want to risk another difficult situation only a couple of weeks later. The upshot is that while a shutdown would be unnecessarily disruptive, it might actually ease passage of a debt limit increase."
One way a shutdown makes the passage of a debt limit increase easier is that it can persuade outside actors to come off the sidelines and begin pressuring the Republican Party to cut a deal. One problem in the politics of the fiscal fight so far is that business leaders, Wall Street, voters and even many pundits have been assuming that Republicans and Democrats will argue and carp and complain but work all this out before the government closes down or defaults. A shutdown will prove that comforting notion wrong, and those groups will begin exerting real political pressure to force a resolution before a default happens.
It's worth noting, for the record, that it would be vastly better if there was no shutdown and no default and House Republicans stopped trying to enact an agenda that lost at the polls by threatening the country. But American politics is what it is right now, and given its sorry condition, a shutdown might be the best of very bad options.