Plane Facts (SPOILERS)
As an Airline Captain, I wanted to add my two cents about technical details about the film. If you would like to accept the film as is, you might not want to read any further! :-)
When it comes to the initial incident, if you have a two-engine aircraft, and you have already shut down one engine, you would not shut down the second engine. It doesn't matter if it is on fire, you keep it running and get as much thrust out of it as you can. Because, with a transport-category aircraft like that one, you are going to go down fast with no thrust. So, Rod Taylor shutting down the second engine would not be correct procedure.
When it comes to the recreation of the incident, Glenn Ford has shut down the right engine, and did not fire the extinguisher bottle. So, that engine can be restarted. To note, the extinguisher bottle fires extinguisher across the engine. This puts out the fire, but will also render it unable to be restarted, because the extinguisher makes it impossible to create proper compustion again.
When Ford shuts down the left engine, he fires the bottle. So, when he asks for the engine to be restarted, it should not restart. In the movie, it does.