When you attach eagles to Eagle of misty mountain facedown they should count as facedown attachments? Cose his game text say:for each card attach….

So if the encounter card make you discard attachment you control can you choose and discard one of the eagle attached to EOMM?

Yes.

They are attachments, and you control all of the cards you bring to the game unless another player/encounter card specifically dictates a change of control. You can definitely discard the facedown attachments to satisfy effects like that on Black Uruks.