Kripke offers a further little intuitive test for telling whether a term is rigid:  Try the term in the sentence frame, "N might not have been N."  If we plug in, for "N," a description like "the President of the U.S. in 1970," we obtain "The President of the U.S. in 1970 might not have been the President of the U.S. in 1970"; and the latter sentence is clearly true, at least on its most natural reading: The person who was President in 1970 might not have been President then (or at any other time).  The truth of that sentence shows the description to refer to different people in different worlds, hence to be flaccid.
    But if we put in the proper name "Nixon," we get "Nixon might not have been Nixon," at best a very strange sentence.  It might mean that Nixon might not have existed at all, which is perhaps the most obvious way in which Nixon could have failed to be Nixon.  But given that Nixon existed, how could he have failed to be Nixon?  He could have failed to be named "Nixon," but that is not to have failed to be Nixon, himself (because, of course, Nixon need not have been named "Nixon").  He could have failed to have the properties stereotypically associated with Nixon, hence failed to "be Nixon" in the same sense in which Dan Quayle allegedly failed to "be Jack Kennedy," but as Kripke says, such flaccid uses of names are unusual.
    Kripke argues that when one uses the name "Nixon" to refer a person in this world and then starts describing hypothetical scenarios or alternative possible worlds, continuing to use the name, one is talking about the same person.  So (again) if you ask, "Might Nixon have joined the Black Panthers rather than becoming President?," the answer may be yes or may be no depending on your modal intuitions, but the scenario you are considering is one in which Nixon, that very person, is a Black Panther -- not one in which whoever or whatever was U.S. President was a Panther.  You are not imagining a world in which a Black Panther is President of the U.S.  (Though, as Ben acutely observed, it is open to a Donnellanizing Russellian to say that you are imagining a world in which a Black Panther is the person who was the actual President of the U.S. in 1970.)