Is an egg classified based on the species inside it, or by the species that laid it?
Species are generally defined in terms of populations (see e.g. the wikipedia page), and it is therefore relatively meaningless to talk about individuals as species. That species is defined in terms of populations is true for many species concepts, e.g. as groups that can produce fertile offspring (biological species concept) or as a evolutionary distinct lineage (phylogenetic species concept). The question is therefore somewhat ill-posed, since it is only framed in terms of individual parents and offspring.

A possible exception could be speciation in apomictic species or speciation events following a change in ploidy level, but this is mostly relevant for certain types of asexual species and probably not what you are after since you are talking about egg-laying animals.