Most trailers the white is ground.
As far as your breakaway, the black goes to the positive post of the battery whatever color wire that is.
Here's a caveat. There should not be a disconnect switch between the brake controller and the positive or negative posts of the battery. The breakaway has to have a constant path to the battery that is never interrupted by a disconnect switch. This allows the switch to do it's job and stop the trailer in an emergency regardless of whether or not you remembered to flip the disconnect switch.
The breakaway does not draw any power until it is engaged (pin is pulled). The warning about disconnecting from the tow vehicle before testing the breakaway is to protect the brake controller. Some brake controllers can be ruined by 12v power feeding back up the brake wire when the emergency breakaway engages.