So let me understand exactly what you mean.
The quad runs for a little while, then dies. You absolutely cannot start it again right away, not matter how many times you kick it. You did check, and it IS getting spark. If you let it sit for a little while, it will start again, and the whole cycle begins again.
Do I have this right?
Try a few things. First, drain all the gas out of it, and fill with new gas. It could be bad gas. Also, make sure your air intake is clear and your filter isn't clogged.
Second, make sure you're getting fuel. If the petcock or the float valve is not flowing properly (but still flowing a little), the quad will run the float bowl dry, and then die. Then, if you let it sit, the float bowl will slowly fill up. The quad starts, runs the float bowl dry again, then dies again.
Possible problems. Try those, if it doesn't fix it, get back to us.
Duncan Racing 340PV, +2 A-arms, +4 LSR axle, and a host of other goodies.