No, if you want a Pokémon to re-learn an Egg Move, it has to have learnt the Egg Move in the first place hence the name "Move Reminder".
And yes you can re-learn a Pokémon an Egg Move, if they have learnt it before and forgotten it.
>In Generation VI, any Pokémon bred or obtained in a Generation VI game may also be taught any move it knew when it hatched or was obtained. This includes Egg moves, Volt Tackle bred onto Pichu, and event-exclusive moves such as the Pokémon Center Tokyo Bay Inkay's Happy Hour.
Source: Experience and Bulbapedia