You can catch everything that is printed through System.out using System.setOut like this:
import java.io.*;
class SystemOutLogging {
    public static void main(String[] args) throws IOException,
                                                  ClassNotFoundException {
        final PrintStream original = System.out;
        System.setOut(new PrintStream("programlog.txt") {
            public void println(String str) {
                process(str + "\n");
            }
            public void print(String str) {
                process(str);
            }
            private void process(String str) {
                // Fill some JEditorPane
                original.println("Program printed: \"" + str + "\"");
            }
        });
        System.out.print("Hello ");
        System.out.println(" World");
    }
}
Prints:
Program printed: "Hello "
Program printed: " World
"
(There is a System.setErr and System.setIn that works similarly.)
If you want to catch stuff that the "subprogram" prints through System.out.println you're in trouble, because System.out is a static so if you launch multiple "subprograms" you'll end up with a mess (since you can't hand a separate System class to each subprogram).
In a situation like this, I honestly think it would be a better idea to launch a separate process through ProcessBuilder. The standard input / output streams of the resulting process could easily be logged.
(p.s. When I think about it, you could probably check the current thread group in the println implementation, and from that decide which subprogram that actually invoked the println method)