In Crystal the day care man gives you an egg with a random baby Pokémon that has high chance of being shiny.
In the Hoenn games, RSE,ORAS there is a Wynaut egg you can get at lavaridge town.
In FRLG, there is a Togepi egg somewhere I forgot.
In Platnium I think Cynthia gives you a Togepi egg in Eterna City, not in DP tho.
In DP tho, a hiker gives you an egg with Happiny, but not in Pt.
In DPPt Riley gives you the Riolu egg in iron island.
In BW, there is a Larvesta egg available in a house on route 18
In BW2 you can get a Happiny egg from a Pokémon breeder somewhere.
X/Y has barely any gift Pokémon compared to previous games and has no egg Pokémon gifts.
In Gen 7, a Pokémon breeder at Paniola Ranch gives you an Eevee Egg.
Also I think you can trade them.