Sun and Moon have significantly changed the breeding mechanic. In ORAS, I could save before picking up eggs, hatch them, then soft reset (SR) back to the save, and get a completely new set of eggs (i.e. the egg was determined at the point you collected it from the daycare - more specifically, it's determined at the point you collect the egg before that one).
However, in Sun and Moon, egg statistics (stats/IVs/gender etc) are determined at some point before you pick up the egg from the Daycare. I've saved, collected and hatched 5 Pokémon, SR'd, then collected and hatched the exact same Pokémon, down to their IVs.
I much prefer this method as it means you don't fill up countless boxes with the same inadequate Pokémon in the quest to find one with perfect IVs and Nature (same applies for Masuda Method if shinies are your thing).
See title for the resultant question.