To terminate a Java program when a user clicks "File >>> Close," the recommended approach is to implement a WindowListener that prompts the user with a JOptionPane for confirmation. If the user confirms, the program can safely close any open streams and then call System.exit(0) to terminate the application. It's important to ensure that the correct instance of any resources, like BufferedReader, is closed to avoid errors. Setting the default close operation to JFrame.EXIT_ON_CLOSE will close all frames, so a custom close operation may be necessary to manage multiple frames properly. Proper handling of window events and resource management is crucial for a smooth application shutdown.