Pappygt,
I think this is going to be easy since the colors you have on the ignition switch are textbook standard.
Your ignition switch is actually two separate switches ganged together on a common turn shaft. Each switch has a separate function as follows:
Switch 1 (Red and Black wires): When the ignition switch is turned to the "on" position these two wires are shorted together. When the switch is in the "off" position these wires are not connected to each other. The red wire goes back to the battery + terminal through the main fuse. The black wire feeds everything that runs off switched 12 volts - the running lights, brake lights (through a brake switch), and the starter motor interlock circuitry. Black is a strange color to use for switched 12 volts, but that is the way it is.
Switch 2 (Black/White and Green wires): When the ignition switch is in the "on" position these wires are not connected. When in the "off" position these wires are shorted together. Note that the connected/disconnected action is backwards from switch #1. The green wire is ground. The black/white wire is the kill switch line. The kill switch line goes to all your kill switches (like the handlebar kill switch, and the tether strap kill switch [if installed]) and to the CDI. When this line is grounded the CDI stops making spark and the engine dies.
So you need switch 1 to start the quad, and switch 2 to stop it. If you decide not to hook up the green and black/white wires you can always use the handlebar kill switch to stop the quad. If you manually short the red and black then you should be able to start the quad, but remember that the battery will always be hooked up to the lighting circuitry until you unhook those wires. If you set the parking brake the brake light will stay on and drain the battery.
Since you've been messing around with the wires be sure to check your main fuse. If you shorted the green and red wires together the fuse will blow instantly, probably without your knowledge.
Rather than use the battery booster box, I would just jump the quad to my car. Just be sure to get the polarity right.
Important Note: If the quad is running (or the engine cranking) there is
high voltage on the kill switch wire (black/white). Mind your fingers. If you touch it while the engine is turning you know it right away. The high voltage comes from the stator through the CDI module and can be a couple hundred volts on a running engine.
I have a different brand 150cc chinese quad. Below is the wiring diagram of my quad. You may find it is identical to yours: