I know of no reason that a civilian would salute the President or anything else. If the National Anthem is played then yes, cross your heart or in some cases salute. This is saluting the flag, not the song or the person. However military folks would need to salute the office of President just as you would if an officer passed by or you came in to report to one. You don’t salute the person (thank goodness), you salute the office or rank. We always had to salute when an officer was present but we saluted the insignia on his/her collar. Same with the president. Look at the button on his collar and salute the office of President.
There seems to be a DOD Reg covering this.
However it does not cover civilians.