I can't see the picture, but I'm assuming that the turtles just hatched for a very brief amount of time when you were not there to apply name tags, so the server butchered them.

@DeejayKingGaming, did you actually see the turtle eggs disappear directly, or is the above scenario possible?

 

The server has a mob butcher that wipes all wild animals frequently to prevent them from overrunning the server with lag, and to let new ones spawn. It's the server itself that is doing this, and applies to both animals that naturally spawn or ones you hatch yourself.

If you tame or give the animal a name via a name tag, however, it will be considered a pet rather than "wild," and thus won't get butchered. You'll need to be there right when the eggs hatch for the best chances of their survival, since the butcher could come very shortly after they spawn in. I recommend waiting at your turtle eggs with name tags if you were to try again.