Has federally funded research at universities and work done at the national labs in the United States been halted due to the government shutdown?
The current shutdown affects federal employees - and even then, not all government departments or agencies are affected. In particular, the Department of Energy, which funds the DOE National Labs, already had an approved spending bill, so DOE is unaffected for now. The NIH and Department of Defense, which fund their own institutes and labs, are in the same situation. However, many other institutions, including NIST, NASA labs, and the Smithsonian have had to cease a lot of their work. Some activities are considered mission critical, and can continue, but that's getting into the specifics. See e.g. this Washington Post article.
University researchers are not directly affected, since they are not federal employees*. Even public universities in the US tend to be on the state level. However, there can be a large indirect effect since e.g. NSF, a large funding agency, has shut down. If this shutdown is anything like the 2013 one, this will not only delay grant applications, but also grant decisions, causing a lot of uncertainty, hiring delays, and lost research productivity.
*The military academies might be an exception, but since the Department of Defense had its spending bill approved they probably wouldn't be affected anyway.
Do researchers just stay at home and not show up to work? Or do they show up, and do something similar to what you would see at a labor strike, e.g. megaphones, demonstrations, make demands?
Generally the former - at least at the largest of the affected places security would (rather sensibly) be considered a mission critical operation, and security would be instructed to stop people from going to work. There tends to be some protests, e.g. at the White House during the shutdowns, but I doubt these are dominated by researchers. There are many, many others affected - some 800 000 government employees in total, many who'd be hit harder than your average federally employed researcher.