The OP is getting a click when the key is turned to start, so his problem may not be the neutral safety switch.
Something to remember about the "neutral safety switch" in the JK:
There isn't one.
There is a bank of 4 switches in the transmission range sensor, inside the transmission. The computer can tell if the transmission lever is in Park or Neutral (or any other position) because the switches are in a certain on-off combination.
If the engine starts in Park and Neutral only, all is well.
If the engine starts in only Park or Neutral, but not both positions, the shift lever cable is out of adjustment.
If the engine starts in any position other than Park or Neutral, or doesn't start at all, the Transmission Range Sensor may be faulty.
Of course, this assumes not even a click when the key is turned to Start.