So yes, windmills will come to a halt because they are designed to. Or, more specifically, they will feather their blades and continue to rotate idly while not producing power in order to ensure adequate hub lubrication.
There is a significant concern that, even in a scenario where all power plants are undamaged, it would take a significant period of time for plant operators and distribution system operators to restore service due to the need to more or less restart everything. Something very similar happened during the California power crisis and the great Northeast blackout, in which undamaged power plants were offline for an extended period of time due to the difficulty of restoring grid condition to normal.
Even under normal conditions of a stable grid, when power plants start (e.g. due to expected increase in demand) it takes hours from startup to connection to the grid. Most renewable generation methods (except hydro) and the newest natural-gas turbine plants can start much more quickly (e.g. 15 minutes), but this assumes that they will be connected to a grid which is operating at their design frequency, which would be very difficult to achieve if there has been a significant loss of generation.
And just to worry a bit more, a significant amount of electrical generation, especially in plants capable of load following (quick response to demand), is via natural gas. Natural gas distribution also relies heavily on compressor stations which run on, well, electricity. These stations have backup power capability but recent experience in the Southwest, during the 2011 winter, shows that this is not entirely reliable, and that severe events can disrupt the natural gas supply as well.
Windmills that e.g. farmers and other rural people install on their property would probably be OK. It should be easy to disconnect them from the grid and have them power local equipment. But without some sort of energy storage, it would be dangerous to try to operate on just windmill power. Too variable. Most equipment wouldn't deal with it very well.
But there is one other strong positive I forgot to mention. PV power. That's becoming more and more available. Such a massive change in just the last few years.
PV would let pockets of people do well. Also, having something like a Tesla car or Powerwall, coupled with rooftop PV, would be a game changer for those people. A setup like that could do pretty well if decoupled from the external power grid.