I have a Manco talon (chinese) ATV, I bought it, drive it ten miles. It starts and runs as long as the starter button and front brake switch are in the start position, as soon as either is released it quits.

The Starter button is normally closed and the hand brake is normally open (open and closed when starting, respectively). I have replaced the CDI unit, the pickup coil, the starter relay.

The coil checks out, as does the stator, battery is at 12.8 volts. Any ideas on what the problem is?    

sounds like you have a short somewhere, or the brake switch is bad.

it could also be the ignition switch losing power.
...has power when the starter button is pressed but dies when you let off...

check all of the wiring coming to and from the ignition switch.

on any ignition switch you have at least a ground, a power wire to allow power to the starter and a primary power wire that keeps power going to the stator and or coil to keep it running.  you probably have a few more wires as well.
especially if you have remote start or alarm.

seriously check all your connections and fuses, if nothing there check your ignition switch.