A lawyer friend writes:
By the way, an interesting tidbit – the administration not only supported the Chamber, but had Neal Katyal — the Acting SG [Solicitor General, who represents the federal government before the Supreme Court] — argue the case. The SG normally argues only one, maybe two, cases per sitting, and that's reserved for the cases the gov't sees as the most important/prominent.
The reason the administration and the Chamber of Commerce saw this case as so important is that they want to hold back mandatory E-Verify as their main bargaining chip for what they really want out of a "comprehensive" immigration deal — amnesty for the Obama administration and massive increases in imported captive labor ("guestworkers") for the Chamber. The decision makes it more likely Congress will be able to pass such a mandate on its own, not bundled with any amnesty or guestworker provisions. In other words, it's not just a win for enforcement but also a setback for amnesty and increased immigration.