First, if the government shuts down the internet, bad things will happen quickly.
In the US, we use the internet to monitor home medical devices, to keep critical infrastructure systems (traffic, transportation, fire, police, phone service, emergency, etc) up and running, as part of VOIP, as part of security systems, and so on. It’s a critical part of commerce and banking, too.
Take all of that away, and people are going to suffer and die in no time flat.
Now, what will we “do” about it? If you have satellite phone (like Iridium), a dial-up modem, maybe Hughes Net, live near the border, or own a smart phones, you should be able to get online in one way or the other. That’s what is happening in Egypt right now. They have a much more centralized internet provider system than we do, and even they can’t stop everything.
Some of those - dial up modems, smart phones and satellite phones, would be hard to disable without crippling the government itself. But if they managed to do it, you’ve still got more ham radio operators than you could count to broadcast information.
The Internet may get disgusted and shut itself down.