GFCI won't reset, charges neutral when wired up
In your case the neutral wire is broken.  

It is rather unusual for wires  to break by themselves, so most likely the problem is at a termination. We can exclude this location, that leaves the service panel.

The neutral wire isn't becoming magically energized when you connect a GFCI to it. The GFCI uses some electricity for its own functions and returns it to the neutral wire. With the neutral broken there is nowhere for that voltage to go, so it stays on the neutral wire, lifting it to 120 volts. The same thing would happen if you used a regular receptacle and a night light.

No receptacle will work at this location until you fix the neutral.