Since mesh networks still ultimately connect back to an ISP, shutdown will still affect the network, slowing communications down. However, as Dibbell noted, the mesh "shadow" network would continue to route information around primary hubs. At worst it would be like an Internet brownout, instead of a blackout. The more nodes in the mesh network, the harder it will be to kill.
Mesh networks aren't an easy answer, either, because they require regular folks to become technogeeks. They also pose some interesting privacy and legal questions. But the beauty is that all kinds of hardware, including smart phones, could be equipped with emergency mesh networking capabilities.
Back in the dawn of the dial-up era, I read an article in Wired about Serbian student activists using smuggled equipment, shaky Internet access and mirror sites to fight the Milosevic regime and get their stories out. Inspired, I emailed myself, starting new friendships that continue to this day as well as a new respect for the power of DIY technology.