Would it be possible to have rank and file coordinates? It would be easier to convey a move and record it as well.
Eagle chess: BISHOP_e3 vs. JohnHS

They are the almost same coordinates as a regular board, but I can add them if you like.
Basically,
8
7
6
5
4
3
2
1a b c d e f g h i

When I capture my piece or pawn with the Eagle, I have to wait to drop the captured piece or pawn on another turn--correct? (I assume I can delay the drop to another move in the future).
One last question....
May a pawn promote to a King or Eagle?
Thanks.

A pawn may promote to an Eagle but not a King. It is your move, btw.
Also, I forgot to add that pieces captured by the Eagle may not remain off the board for more than two moves if legally possible.
E.g. An eagle captures a rook. If the rook is not dropped on the board next turn, then it must do so next time if legal. If the king is in check, it must be used to block the check if possible. If the king is repeatedly checked and there is no way to block the checks by dropping the rook, it may stay out of the game until it is able to block a check or the checks end.
Two pieces may not be 'carried by the eagle' at the same time.
It is your move, good luck!
Eagle Chess
@Bishop_e3 vs. @JohnHS
RULES: The eagle piece, which starts on i1 and i8, moves like a queen and a knight combined. It may jump over pieces when moving like a knight. It cannot capture enemy pieces, though it may be captured. It may capture any one of its own pieces (including the king), and next move drop it anywhere on the board that is not already occupied by a piece of either color. All other rules the same as standard chess. Time control 2 days per move, 1 day warning period.