There are specific ranges for "sensitive eyes". The most important thing is to thoroughly moisturise the area around the eyes before applying make-up. And to carefully remove eye make-up with gentle products adapted to this part of the face.
Eye shadows have a high pigment content (zinc oxide and titanium oxide) for maximum coverage. This can make them irritating. Mascara and eye-liner may be in contact with the eye and require a specific formulation. They contain waxes and gelling agents.
However, over time allergenic components (rosin, kathon, etc.) have been removed from make-up products. On the other hand, eczema on the eyelids would point to an allergy caused by contact with a nail varnish or an airborne allergen: spray-on perfume, hair lacquer, plant elements or pollen, etc.