I have eczema too. I don't have much problem with sunscreen ingredients but I have a major problem with the sun. Anytime I get any sun exposure, my eczema flares up. I can't even recognize myself, myface becomes very red and flaky.
This is why I purchase sunscreen with the most zinc oxide. Zinc oxide has to be one of the first ingredients listed because this means the product contains more of this ingredient. I also wear a large hat and big sunglasses and generally try to avoid the sun as much as possible.