I am a us citizen based in the us. I would like to register a co.uk domain. Do I need to be a uk citizen or have some other sort of connection to the uk to register that domain?
For instance, .ca (canada) requires this and that is why I ask.
No, there are no restrictions like that for .co.uk domains. You can go to any major registrar and register one if it's available.
If you would like more information this is the official Registry of .uk domains and their rules: http://www.nominet.org.uk/registrants/aboutdomainnames/rules/
Yes, you either have to have citizenship, be a resident or have a UK registered company.