From Bulbapedia, the community-driven Pokémon encyclopedia.
The Odd Egg (Japanese: なぞのタマゴ Odd Egg) is a specific gift Egg the player receives in Pokémon Crystal. The Pokémon it hatches into has a higher-than-usual chance of being Shiny.
The Egg can hatch into Pichu, Cleffa, Igglybuff, Tyrogue, Smoochum, Elekid, or Magby. Regardless of the Pokémon that hatches, it will know Dizzy Punch.
In the Japanese version of Pokémon Crystal, if the player had received an event that distributed the Egg Ticket via the Pokémon Mobile System GB, they could obtain an Egg Ticket from the Day-Care Man at the Pokémon Day Care on Route 34. If taken to the Pokémon Communication Center, which is located in Goldenrod City, the player could exchange it with the Pokémon Center Nurse in the Trade Corner for the Odd Egg.
In the international versions of Pokémon Crystal, the player could receive the Odd Egg from the Day-Care Man at the Pokémon Day Care on Route 34. This change was likely due to the Pokémon Mobile System GB not being a part of the international version of Pokémon Crystal.
Pokémon hatched from the Odd Egg can have one of two different sets of possible IVs (Attack/Defense/Speed/Special): 0/0/0/0 (not Shiny) or 2/10/10/10 (Shiny). Due to gender being determined by IVs in the Generation II games, this means that all hatched Pokémon will always be female (except Tyrogue, which is always male). Additionally, non-Shiny Pokémon hatched from the Odd Egg will always have Grass-type Hidden Power with power 31, while Shiny Pokémon will have Fighting-type Hidden Power with power 49.
Unlike other Pokémon hatched from Eggs, the Pokémon hatched from the Odd Egg have 20 friendship (rather than 120). All Pokémon hatched from the Odd Egg hatch with 125 total experience; while this is the minimum experience for a level 5 Pokémon in the Medium Fast experience group experience group (most possible hatched Pokémon), this results in 25 excess experience for Pokémon in the Fast experience group (Igglybuff and Cleffa), which only need 100 experience to be at level 5.
In the Japanese version of Pokémon Crystal, there is a 50% chance the hatched Pokémon will be Shiny (IVs 2/10/10/10) and a 50% chance it will not (IVs 0/0/0/0).
In the international versions of Pokémon Crystal, there is a 14% chance the hatched Pokémon will be Shiny (IVs 2/10/10/10) and an 86% chance it will not (IVs 0/0/0/0), but the chance of any particular Pokémon species hatching from the Egg and the chance of any given species being Shiny are not uniform.