IMO, shutting down the internet (for, let’s say, 95% of the population) is not a technical challenge, but a political/legal one – which fast becomes irrelevant when the country is ruled by a dictatorship which would presumably be capable of enforcing its will with efficient deadly force (it did take over the US, after all).
Talking about shutting down DNS servers is missing the point; there are a variety of methods of stopping internet access to the masses, and most of them depend on the unfortunate fact that today’s Internet is not the highly-redundant web of peer-to-peer interconnections that the military envisioned. (See Stathtol’s post)
The Internet, as it is known to most Americans today, is primarily run by large advertising, telecom, and electronics companies: Google, Verizon, AT&T, Comcast, the various backbone providers, Apple, Microsoft, etc. Any one of these giant, centralized corporations is easy pickings for a dictatorship; a few beheadings later, you can bet that the rest will fall in line. Whether you do it by switching off servers or power plants, cutting trunk lines, destroying satellites, software updates that cripple equipment, etc. is of little consequence, because sooner or later, without popular armed resistance, you will disable enough equipment simply because there’s nobody powerful enough to stop you.
The point is that no matter how resilient the equipment may or may not be, their flesh-and-blood operators are infinitely more vulnerable. Were the OP’s question “How resilient is the Internet against non-physical cyber attacks (only)”, the situation would be vastly different. Under the proposed dictatorship scenario, little of that matters: How many people would really risk execution/torture/labor camp/etc. for daring to implement technical workarounds even when they’re possible?
The Chinese firewall is a different beast altogether because they must leave parts of the Internet alive while filtering other parts, which is a much, much harder challenge than blocking it altogether.