Why isn't the genative case used here in the negation? Ні, я не їм яєць.
Your translation is now accepted as an alternative. Actually, it should be the best translation, but this construction is currently not taught in the course.
When negation is about something other than a living being, both accusative and genitive cases are usually OK. But yes, genitive sounds like a more educated speech.
Why isn't "Ні, не їм яйця" not accepted? Is Ukrainian not pro-drop?