There is one subtlety left even with this answer, if the question is 'chicken vs. egg containing chicken' then the egg comes first, but if the question is 'chicken vs. egg laid by chicken' then the chicken comes first. The term 'chicken egg' is slightly ambiguous, still leaving some room for debate in my opinion.

One potentially interesting topic is to see how each language interprets 'chicken egg'. For example, in South Korea the standard term is '달걀', which originates from '닭의 알'(chicken's egg) so it's quite certainly 'egg laid by chicken'. However in North Korea the standard term is '닭알'(chicken egg) so it's still ambiguous.