They base the constitutional argument, not on the 14th amendment clause, but on the principle established by Abraham Lincoln that sometimes a president has to violate one law in order to fulfill another, more necessary law. In this case, it would be violating the debt ceiling rules in order to carry out the presidential responsibility to see that the laws are faithfully executed. Here, that would be the obligation to pay our nation's debts.
I leave aside the constitution questions; what interests me more is the political position. Rather than being held hostage by Republican intransigence, this would free Obama from making any more concessions. As Posner and Vermeule point out:
Politically, he can’t lose. The public wants a deal. The threat to act unilaterally will only strengthen his bargaining power if Republicans don’t want to be frozen out; if they defy him, the public will throw their support to the president. Either way, Republicans look like the obstructionists and will pay a price.That makes sense to me.
And, if Obama will only threaten to do it, Posner and Vermeule predict that the Repubs will then adopt the McConnell plan, whereby Congress in effect gives Obama "permission" to do just that, thereby wiggling out of the political hot water of being labeled obstructionists. Either way, Obama wins. It's the best bargaining chip he has right now.
So, just do it, Mr. President. And let the Repubs writhe in the political hot seat they bought when they elected the Tea Party crowd.