As far as I can tell, quest eggs are just like Level 3 eggs from the AP shop, eggs that always successful up to level 3. I have never failed a data inject nor broke it. So, I just assume it's a Level 3 egg. You cannot go over 3/5. I hope this helps.
On the side note, if you scan a quest egg, u get a character bounded data depending on what the egg is (I.E. Kunemon quest egg gives you character bounded insect data). XD