Basically what I'm asking is if an animal of species
x were to lay an egg, and the animal inside that egg happened to be the first member of a genetically new species
y, would the egg be considered an
x egg or
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.
An Ostrich egg. A chicken egg. A quail egg...whether you call an egg based on who laid it or who is inside of it is not a question of science but a question of linguistic.
Non-fertilized chicken eggs do not contain any chicken. We still call these eggs chicken eggs. Following the same logic, I would call an egg based on who laid it. But really this has nothing to do with science.
But, the first question one should ask is whether the scenario you are describing makes any sense. Is it possible for an offspring to be of a different species than its parent?
Can the offspring belong to a different species than the parent?
The concept of species is very poorly defined. The reason is not that we fail to agree on a definition but that the categorization of living things into discrete categories often does not make sense.
Let's think of an individual of species
A. Imagine we were to look back at the ancestry of this individual for a relatively high number of generations. You chose one specific ancestor in the lineage and realize that this ancestor looks very different. Maybe we have already named the species that this individual belongs to. Let's say the ancestor belongs to species
B. So, at some point there was necessarily an individual from species
B who had an offspring who belonged to species
A, right? ...not really. The reason is that the concept of species makes little sense. There is no such thing as absolutely clearly defined groups of individuals that you can gather into one single species.
We often think of the concept of ring species when thinking about population separated spatially. But the same concept hold true for population separated temporally. If we come back to the two individuals belonging to species
A and species
B. These two individuals probably cannot reproduce together and are very different. It seems legitimate to classify them in different species. However, all individuals along the way were probably able to reproduce with their own parent and offspring.
This question is interesting. First of all, I recommend you to look a little bit about the most popular definitions of species and, more importantly, how speciation can occur.
I can picture a problem of hybridization when I think of your question. Since speciation takes many (plentiful) generations to happen, and as I can remember, can involve sympatric speciaton and/or extinction of the parent's species since the new one won't be enormously different in matters of food request, habitat and other physiological needs (generally) and so there will be a conquest/fight for resources (you can look out for more details in Charles Darwin's On the Origin of Species). All this to say you won't have a newton coming out of a frog's egg. :)
But hybridization can lead to speciation, and you can have a hybrid coming out of an egg of a well descripted species. This happens with more frequency in fishes, insects and amphibians. For example, you can study individuals that are phenotipically different from their parents, with other types of adaptation (sometimes better) to the environment, and that emerge from different (and phylogenetically proximate) diploid sexually reproducing species... and the resulting individuals can have, of course, di/tri/tetraploid genomes... So, morphological and genetically different from their parents!!
And when referring to these situations, scientists generally call them genus name of the species involved + complex. For ex, the Pelophylax complex, involving 3 types of frogs.
If you really have an urge to search about this, try to go to this paper.
Hope I helped.
PS: You can also research about the Squalius alburnoides complex, which is about a group fishes extant in the Iberian Peninsula, with an amazing (and complex) process of speciation where you have a well characterized hybrid coming out of eggs layed by hybrids or parent species.