Diamond/Pearl's Pokedex entry for Eggecute notes that, "Its six eggs converse using telepathy. They can quickly gather if they become separated." The eggs are all capable of telepathic communication, so they must be alive. Omega Ruby/Alpha Sapphire's Pokedex entry also notes, "When cracks increasingly appear on the eggs, Exeggcute is close to evolution," so the egg in the back may simply be an indicator that the Pokemon is ready to evolve.