As Mike Manes of EOSS suggested, I ran the geiger counter in a vacuum to make sure it didn't arc over. The tube has 600 volts across it, so this is an issue in low air pressure conditions. At first it looked fine. Then at 28 inches of mercury vacuum (about 94% vacuum), the LED indicator remained on. Some where the 600 volts was finding a ground and triggering the LED. I started coating the circuit in hot glue and have it running at lower pressures, but stil not in a full vacuum. I ran out of hot glue to do any more work, so I'll work again in this tomorrow.