Even if the spell or ability allowed you to choose a card "you own ... in exile" (e.g., Coax from the Blind Eternities speaks of an "Eldrazi card you own from outside the game or in exile"), the card would still have to meet the criteria of that spell or ability. Note, however, that in general, a face-down card in exile has no characteristics, even if—
the card would meet the spell's or ability's criteria if it were face up, or
something allows one or more players to look at that card in exile
(C.R. 406.3a).
Thus, for example, in general, a face-down card in exile isn't an "Eldrazi card" for the purposes of Coax from the Blind Eternities, since the face-down card has no subtypes, including Eldrazi (C.R. 203.5m, 109.3).
EDIT (Feb. 4, 2021): Edited to conform to rule update for Kaldheim.
- the card would meet the spell's or ability's criteria if it were face up, or
- something allows one or more players to look at that card in exile
(C.R. 406.3a).Thus, for example, in general, a face-down card in exile isn't an "Eldrazi card" for the purposes of Coax from the Blind Eternities, since the face-down card has no subtypes, including Eldrazi (C.R. 203.5m, 109.3).
EDIT (Feb. 4, 2021): Edited to conform to rule update for Kaldheim.