What causes my chest and back to feel like someone just shot my body with a thousand needles when I sneeze?

Nerve pain. Probably nerve pain due to bulging disk in neck or upper back or early shinlges if you have a rash on one side only.
? nerve. One theory: if you have an underlying neck or back problem such as a herniated disc, sudden movements such as sneezing or coughing can potentially cause the disc to exert pressure at the nerve and cause pain. Another thought: if you are experiencing concurrent cold symptoms source of pain could be pulmonary. Talk to your doctor regardless.