Here's a quote from This link
"Incidentally, all grounding of house wiring must go directly to the "grounding bus" in the electrical panel. Older techniques, including attaching a ground to a cold water pipe or to a rod in the earth are no longer acceptable for individual circuits."
My box doesn't seem to have a "grounding bus". The grounding bus and where you connect the white wires appears to be one and the same. The only real solution is to replace the circuit breaker box.