Will vitamin/mineral deficiency cause eczema-like symptoms? (if yes, what vitamin (s) or mineral (s)?)

Agree with Dr. R. Dr. R has listed the causes of nutritional skin rashes, however, if you are eating a regular mixed diet, nutritional deficiency is an unlikely cause of the rash. Allergies are a common cause. If the rash has lasted more than a couple of weeks, it would be prudent to see a doctor.
Yes. Low zinc levels, low magnesium levels as well a deficiencies in vitamins c, b2, and b6. Along with low essential fatty acids can be associated to eczema. Do not forget the strong hereditary and allergic component.