Why does one leg of my panel drop to 0v and the other rise to 240v when a load is connected?
It sounds like you have a loose, broken, or disconnected neutral at some point.

Double check all terminations are tight, and contact the conductor, not the insulation. 

Also check that you have the correct earthing/bonding methods. Do you have an earth/ground wire back to the house?

If so, you need a separate earth and neutral bar, with each wire connected to the correct bar, and no link between them.

If not, you need an earth electrode connected to the earth bar, the incoming neutral connected to the neutral bar, and a link between the two.*

* I believe the NEC no longer allows this approach; I'm not familiar with US code. It may be grandfathered in.