If you want to hatch a Pokemon with the same nature, use an Everstone.
As for Synchronize,
Starting in Pokémon Omega Ruby and Alpha Sapphire, if a Pokémon with Synchronize is leading the party, gift Pokémon (except Pokémon obtained via Mystery Gift) that are in the Undiscovered Egg Group will always have the same Nature as the Pokémon with this Ability. Starting in Pokémon Ultra Sun and Ultra Moon, this applies regardless of the Pokémon's Egg Group; however, this does not apply to the Eevee Egg received at the Pokémon Nursery.
Bulbapedia article on Synchronize