Recording any footage in Hawknet Freebuild is fine, as long as you fully read and agree to our Terms and Conditions. Of course, since you've been playing on the server, you should have read and agreed to them while playing.
You can use Hawknet's name in your video, and state which server you are playing on, however you must not make any claims for the name or server under your name. Any queries or comments from users stating what server it is, or what its name is, you must direct them to the server, or these forums.
Other than that, and what's been mentioned in other forum posts regarding youtube (try the search feature), you're free to record footage in Hawknet Freebuild.