Theoretically a country has the right under its sovereignty to use whatever currency it likes. Trading with other countries might be a small issue but it wouldn't be insurmountable.
The one issue I think right now most countries would have with the idea is that cash is often used in secret government projects and law enforcement's use of cash such as undercover investigations and as evidence in charges dealing with bribes. Since cryptocurrency involves open ledgers that are public then there is a slight chance a savvy criminal might deduce the cryptocurrency is connected to law enforcement. Serial numbers on dollar bills don't have that kind of liability. This would put the lives of undercover law enforcement officers at risk.