Note that oviraptors do NOT collect the eggs of your (tribe's) tames. Oviraptors are useful because they will collect wild eggs and will allow you to pick them up without aggro-ing any nearby dinos.

To use them, set them to passive and following you/your flyer, drop them near the egg, fly closer to it, then pick your oviraptor up after it snatches the egg. Fly to a safe place (if you're in dangerous territory) to collect the egg!