I've read various explanations why as many as 22 reactors shut down during the blackout. I haven't verified the number, but I know why they shut down. Because of Federal regulations.
This isn't to question the Federal regulations in this regard - they're heavy-handed, but I don't have any better ideas. And it shows just how much of a safety margin we have with the current generation of nuclear power plants.
It's been a few years, so I don't recall all the details, and these will vary from plant to plant as described in a licensing document known as the "technical specifications". If I recall correctly, a plant had to be in cold shutdown within a fairly short time after the loss of offsite power. Cold shutdown means basically that the plant is not "critical" (no sustained nuclear reaction is taking place) and is depressurized, and essentially is in a state where it requires the minimal amount of effort and power to control it. This is as opposed to a "hot shutdown", in which the reactor isn't delivering any steam to the turbine, but would be ready to if you opened the valves isolating the reactor building.
It's much like the distinction between having your car shut off and having it idle. Except that startup on a nuclear power plant is a much more time-consuming process. Part of this is because it takes time to warm large metal objects several hundred degrees without causing thermal distortion or other problems, and these problems are also found at fossil powered plants. Others are caused by the presence of components unique to nuclear power plants, such as hydrogen recombiners and a boiling water reactor's offgas system. The rest is caused by regulations. It doesn't take too long to come back up after a hot shutdown, but it'll take a day or so to come back up after a cold shutdown.
So why does a nuclear power plant need offsite power? Actually it doesn't, any more than you absolutely need to have a fire extinguisher. Again, regulations require a combination of batteries and other forms of stored energy (diesel fuel and compressed air, for instance) to make sure that the plant can deliver enough power by itself to maintain cold shutdown conditions and support emergency loads. What's more, there are redundant supplies - either of two huge generators per unit can do the job by themselves.
One example was a TVA plant that was never completed, with two 1300MW reactors. It had four 16 cylinder turbocharged diesel generators, (two per unit), each of which was designed to provide about 7.5MW. They were started with compressed air, and enough was stored to start them multiple times. For fuel, they had 4 tanks each embedded in concrete which were on the order of 10 feet in diameter and at least 60' long, and a specified amount of fuel was available in all of them at all times when they were considered available. Because of the compressed air and battery power, the diesels could be started without any offsite power available.
How big is such a beast? The cylinders were 17" across - most of you reading could get inside them.
How much is 7.5MW? Let's say you used 1000 kWh on your last monthly electric bill. A generator that size generates that much energy in 8 minutes, and there were two of them per unit.
If I recall correctly, the source of all these regulations is Title 10 of the Code of Federal Regulations - see part of it here.