I believe in Roger's case at JFK, debris from the failed engine damaged the adjacent engine's fire detection system causing a false fire warning, rather than a second engine failure as such.
If I remember correctly I believe he realised this and in view of the critical nature of the event (max weight, high air temperature), kept that engine running until they got down safely.
One of those fine examples where, in an emergency, it may be necessary to break all the rules and live to argue your case at the subsequent enquiry from the safety of an office chair ....