The back up lights on my '87 240 w/M47 stopped working a long time ago. Several months later I was backing up at night and had to wait for a car to go by, and then I noticed two white reflections coming from the car. My back up lights had returned! I have no idea what the problem was, but I didn't do anything and they came back and have worked ever since (at least 6 months). The circuit cards in the taillights are easily worn, so if you see silver on the copper section, that may be the problem (but unlikely that both would go out at the same time). Have someone put the car in reverse while you jiggle the lights to see if you can get them to come on. Every so often I have a taillight go out, and moving the bulb holder usually fixes it. If that doesn't work, then I'd bet on the switch.

Good luck.