Breast sonogram reported cyst, should I push draining it?

No need to. A cyst is completely benign and do not need to be drained. If it is causing you pain or discomfort you can ask to have it drained. Placing a needle through the skin to drain it under ultrasound can be done simply and with little risk, but you can always introduce an infection or hit a blood vessel. Why take any risk if no benefit.
No. Absolutley not. Forget about it. If you drain it it will come back.