Should I have current running through my neutral wire in breaker box?
Is this on all your circuits?  Are half your circuits under 120V while the other half are over 120V by about the same amount?
If so, turn your main breaker off IMMEDIATELY. Call your power company and report an OUTAGE.
It actually is a genuine outage.  One of the wires from the pole has broken.*  It's a 30-minute fix for a power company boom truck, happens all the time, they know what to do. Nobody else can do it.
Call the emergency outage number and report your power is out. **
When it happened to us, they had somebody out in an hour on a Sunday.
Fix it right away.  Don't fool around with it.
What's happening is that without a neutral wire, your voltages are a &quot;teeter-totter&quot;.  It's like all the 120V loads on one leg are on a teeter-totter with all the 120V loads on the other leg.  When the loads are inbalanced, one goes down, the other goes up, and then it destroys appliances that are only made for 120V.
This teeter-totter changes instantly based on load. The coffeemaker auto-starts, and boom!
When it hit our complex, several people had their microwaves and refrigerators destroyed.


* Because of the weird way North America does electrical power, all single wire breaks don't seem like outages.  Both legs will work but with weird voltages.  Above I describe how lost neutrals behave.  When a hot wire is lost, the dead leg will still work (weakly) because power is backfeeding through 240V loads like water heater.    Some people observed their lights come back on when they turn on their oven.  That makes sense.
** DON'T go so far as mentioning that the power is out because you turned off the main breaker :)  We've had reports of service agents telling people that because their lights partially work, they don't have an outage.  That's wrong. The agents don't understand what I just said above.