Will aspirating breast cyst show cancer cells or I'm just being paranoid?

Likely benign. Most cysts in breast are benign. You are prudent to have the abnormality checked out.
Explain. Your anxiety to your doctor. They may aspirate the cyst to give you peace of mind. Reasonable request on your part.
Almost-never. Breast cysts are benign fluid-filled lumps that develop due to blockage within the milk ducts. A mammogram cannot differentiate between a tumor or cyst, but an ultrasound can. Rarely, a cyst can look atypical on ultrasound, leading to the recommendation for aspiration. Unless the cyst fluid is bloody, it is recommended to simply discard this fluid rather than test it.