Each monster's egg, called spores in the files, is uniquely identifiable, even before the egg hatches. Since each monster in the game is also made from a unique combination of elements, these eggs also represent the 'genetic material' of the two parents shown at the top of the breeding structure. Kayna initially did not have an egg; its egg was added in Update 2.2.4.
All Rare monsters and Epic monsters had the same eggs as their non-Rare/non-Epic counterparts, except that a shimmer or gleam effect will appear across the egg from left to right in Breeding Structures and Nurseries, but on September 25th, 2019, all Rares and Epics got their own unique eggs! All Rare and Epic Monsters got their own Eggs, no matter what class they had.