No Tom, The Black and White is for the Kill side of the switch.
The Red and Brown is the one to test.
Pull apart connector, Use 12V meter, test from battery ground to the Red wire on the ebox side and make sure you have 12V battery power. Then move the ground lead over to Black at the connector and test again, this will tell you if you have a bad ground on the ebox side, Which is by the way is the Black wire for the stator going to the engine, this is where the ebox gets it's ground from.
Now set the meter for ohms and check for continuity on the switch side of the connector when pushing the start switch. If no contact it won't activate the relay.
A simple check is to use a some needle nose pliers and short together the Red and Brown on the ebox side and see if engine cranks over, if it does then a bad connector or switch, if nothing happens then a possible bad relay or connection inside the ebox.