If the term "fat and happy" were to be taken literally, the opposite statement would be "thin and unhappy." While this might overly simplify both mental state and eating habits, it does -- in part -- ring true. Depression notoriously leads to reduced appetite and is a top cause of unintended weight loss [source: MedlinePlus].
Loss of appetite is a neurovegetative symptom originating in the autonomic nervous system [source: Dantzer, et al]. It keeps a person from finding pleasure in things they once enjoyed, and eating is a source of pleasure for most people.
There is an exception to this behavior, however. People with atypical depression tend to eat more and can actually experience weight gain [source: WebMD].