Yes, that is not true. Originally as one of many stories goes, the US government originally formed the internet by allowing some local school systems connect to their databases for research and informational purposes. Then, once it got outside their control, they tried to shut it down by turning off their servers, however, that didn't work and that's one story. Maybe true, maybe not, but the point is that we would have to have a world wide power outage and communication disruption to shut down the internet, and even then, I doubt it really would. Probably your friend, assuming they live in the US, meant that the US government has copies of everything to monitor, but even that's extremely unlikely, unless they have some means of storage that hasn't even been fathomed by the general public yet...
