There is a difference between Raynaud's disease, which is not known to be precipitated by any particular medical condition, vs. Raynaud's syndrome in which the Raynaud's phenomenon of fingers and toes turning white, then blue then red, is associated with several conditions, some bieng connective tissue diseases. One particular associated condition, cryoglobulinemia might have serious exacerbations if a patient with this disease had CoolSculpting.Although most Raynaud's disease patients have the phenomenon occur when they're exposed to cold ambient temperature or cold temperature applied directly to their skin, it is doubtful that freezing the abdominal subcutaneous fat will cool off the core temperature of warm blooded human beings enough to cool the finger tips, toes, nose and earlobes. However, the Zeltiq company certainly didn't test this, nor apply to the FDA for clearance in such individuals. It would be a risk for anyone with Raynaud's to have CoolSculpting, and I would not recommend it, unless definitive scientific testing were done proving the safety.
The information provided in Dr. Shelton's answer is for educational purposes only and is not intended to constitute medical advice. The information provided should not be relied upon as a substitute for consultations with a qualified health professional who may be familiar with your individual medical needs.
Web reference: http://www.thenyac.com/zeltiq/index.html