One of zygovo's mechanics is that it is a specialist egg eater, specifically it has been said that they are able to eat player's eggs. With other similar pvp mechanics (hatchling carry, BW horn attack, etc), in pvp players are usually required to be grouped before being able to interact with other players. This could work for player egg eating, but with multiple eggs, there is a risk that the zygovo could take an egg that the parents wanted to keep.
It would be great if the nesting menu had an "abandon egg" option. In pve, zygovos could only eat eggs that were abandoned in this way by their parents. It could be visibly identified as an egg that has rolled out of the nest. In pvp, abandoned eggs would be guaranteed to be picked up first by a zygovo, if the parents are nearby and guarding the nest. If the parents aren't close enough nearby... Then the zygovo picks up a random (or even selectable) egg from the nest, not just abandoned eggs