How to use Elite Eagle Warriors?

Elite Eagle Warriors are probably the best unit in the game, if we were to rank all the units on how well they win games for you. They play a very specific but game changing role for the Meso civs.

Strategies

Don't bother with Eagles earlier than Imperial.

But once you hit Imperial, research Elite Eagle and the right techs and start pumping them out. When I say "right techs," I mean Blast Furnace and/or Garland Wars, Plate Mail Armor and/or El Dorado, Squires if you haven't yet already

Use Elite Eagles to raid your enemy's economy. They are highly resistant to arrow fire with 8 pierce armor with all Blacksmith upgrades, and can kill Villagers without much impunity. They are also strong enough to destroy Town Centers, which can help in a pinch. You should always try to support them with Rams, and Arbalests if you can afford to.

The reason why Eagles are so effective is their cost. They are very cheap at 20f and 50g a pop. When you hit Imperial, you will be researching all kinds of Food heavy techs so the Eagles being so Food cheap makes them very easy to mass. The Aztecs in particular have a "cheap" Unique Tech at 450f and 750g that gives Eagles +4 Atk.

The Elite Eagle Warrior upgrade is also very, very quick to research. You can have fully upgraded Eagles long before your opponents can field anything siimlar.

Counters

Another reason why Elite Eagles are so strong is that they are difficult to counter, at least at first.

Paladins counter them. Cavaliers? Not so much.

Champions counter them but Two-Handed Swordsmen don't fare as well vs them.

You've probably noticed that those two units need TWO upgrades to get to their strongest forms, and that's alot of time and resources to spend. If you play your cards right as a Mayan or an Aztec you will have the game well in your favor by the time your opponent can pull out Paladins or Champs.

Eagles themselves counter any and all ranged units, whether they are Archers, Siege, or Monks. Hand Cannoneers and Heavy Scorpions can give them a tough time because of their high attack that can overcome Eagle Warriors' pierce armor...if they have a meatshield. However which units make the best meatshields? Champs and Paladins.

Furthermore, Chemistry has one of the longest researching times in the game and Heavy Scorpion is almost unresearchable in early Imperial at 1100f and 1000w.

Some UUs are fairly quick to mass up and counter Eagles humbly. Generally Infantry UUs will beat up Eagles, but the most important one you should watch out for are Huskarls, who can come from Barracks and will ruin your economy just as well as Eagles can ruin their enemies.

If your enemies survive your initial onslaught, then the aforementioned units will counter Eagles. Paladins and Champions with Hand Cannoneers or Heavy Scorps backing them up, and then Woad Raiders, Berserks, Throwing Axemen, Teutonic Knights, Cataphracts, War Elephants, Samurai, and Jaguar Warriors.

Bonus

Aztecs create Elite Eagles +15% faster than the Mayans (or the Inca if you play FE), and Garland Wars gives them +4 Attack. Aztec Eagles are the strongest early Imperial, but are not nearly as useful after that. They still make a good archer counter but they are quite easy to beat.

Mayan Eagles get +40 HPs from their El Dorado Unique Technology but you won't be researching it quite as quickly as the Aztecs. Mayan Eagles are not quite as good as the Aztecs in early Imperial but they are definitely more useful if you have to put down an enemy who survives the early onslaught.

A Goth ally will allow you to create Barracks units, of which Eagles are one, +20% faster.

In addition to what the battler said. During early castle with the Mayans you can do an eagle rush which is an extraordinarily effective strategy. Take this and transition to Elite Eagles during Imp and El Dorado makes their Eagles more effective RAIDERS than the Aztecs.