Since there is not yet a "hello" or "goodbye" emote, saying goodbye for an eagle is extremely tricky. Although it can simply note that it is sad and then leave, this seems to be an unofficial goodbye and leaves their friend feeling a bit stumped.
However, there are some solutions!
Here are some things eagles can do to say goodbye:
1. Trying getting your friend's attention by using the sad emote. Then, put down the "X" emote along with the group emote, and repeat the sad symbol. Your friend may not exactly understand that your leaving, but they will understand that this is something to be sad about and will say goodbye to you.
If an eagle does this to you an you are a bit confused as what they are trying to say, try putting your own "goodbye" emote a long with a question mark, to ask "are you leaving?". If the eagle is leaving, it will put a "yes" emote.
2. Another thing an eagle can do to say goodbye is to put down the sad emote along with the eagle's smiling emote, which also means hello in the eagle language. This may be a bit more confusing, however.
3. The Snuggle Kitten77 suggested this one: you can also show the "X" or "no" command, followed by the "wait" command, then express sadness afterwards. This means that the eagle is out of time.

This emote has a closed smile like a regular animal's "hello" emote would have, but does not have the familiar wave of the paw/claw/hoof. Because of this, the emote is often used for both purposes (also because eagles do not have have the "hello" emote). When meeting a player, an eagle will use this emote, and will use it when they are happy as well. It is essential to understand this as it can be placed alongside many other emotes to form a complex sentence or feeling.
In the RP server, an eagle may use this emote to show mischievousness.

Eagles lack a emote of them sticking their tongue out, being silly. There is a way that eagles can show this.
A simple showing of the eagle's derpy confused emote along with the smiling emote can be used to express silliness. Also, any emote followed by the smiling one can express silliness or that the eagle is fooling around.

Eagles have a hard time in dense forests when they have a friend or a group they are leading. When flying low is necessary, eagle's wings are easily caught on branches an leaves, as well as two trees that may be right next to each other. If you are running around with your eagle friend, and for whatever reason they stop, this may be because they are stuck behind some trees. Be patient, as they will soon get out and you two will soon continue on your way. Fog is also a problem, because an eagle looking for obelisks and crystals in that area will suddenly have a hard time seeing, when flying in the air. They may also loose the group! Don't be frustrated if an eagle takes a long time looking for obelisks, as the possible fog may be hindering that ability. Another eagle hazard are clouds. Although only common in the mountains or way up high, they present a dizzying and frustrating hardship, mostly because they appear out of nowhere and render the eagle's sight worthless. Just wait, as soon the cloud will go away or the eagle will make its way out. Also, note that eagles can't fit through most cavern entrances. If it is absolutely nessicary that you enter into a cave, you'll have to leave your eagle friend behind. (Unless they meet you at the end, if it's a shorter cave).

Eagles do several things that are key to helping groups find obelisks and crystals. These things are important to know if you want to find all of the things you're looking for.
If an eagle is leading you to obelisks and crystals, chances are it going to, out of no where, flying high up in the sky until you can't see it anymore. Don't worry- the eagle is getting an eagle's eye view (no pun intended) on the surrounding area, checking for obelisks and other goodies. Simply stay where you are, even if the eagle flies off, as once it has found something it will come back to you and lead you to it.
If you are in a fairly large group and suddenly you hear the spamming of the eagle's screech, then that means that the eagle found something and is trying to get the group's attention, or has to say something important. Alert the rest of the group and find out what the eagle is trying to say.
If you are with an eagle and it suddenly keeps rising and diving, over and over again, then that means it is just trying to pick up speed. Rising and falling make the eagle faster, so eagle players will use this like any other animal will use running. When an eagle flies without doing this, they are about as fast as a badger cub running.
(Thank you to Kaija for the following suggestion): Eagles can not collect anything that gives essence, like flowers, mushrooms, crystals and fruit. If you see an eagle hovering around any one of these things, then putting down the sad emote, it means that they are dispointed that they can not collect the object.

Some may think that all eagles do is lead big groups to obelisks, but this isn't entirely true. Eagles want to have fun, too, even if there is a limited amount of things they can do.
1. Spinning
It is quite common to see an eagle spinning. They do this to fool around or to show their excitment when seeing another eagle or animal. When an eagle begins to spin for what seems like no apparent reason, try to spin along- it's a way for both you an your friend to have fun!
2. Kicking around the Puff Ball
If you're an expert puff ball player, then prepare to meet your match. Eagles can play as well, and when they dive and kick it, the puff ball goes flying. They'll ruin your game of keep it up in an instant, but hey, it's all in good fun.
3. Mud
If you and an eagle friend come across a mud puddle- well, it's not recommended that you go and jump in it. Eagles can't do that, and they'll probably be sad.
4. Eagle Only things.
Amongst all of these things, eagles also enjoy flying under the map, into the sun and moon, and diving off cliffs. This is something regular animals can't do, so, for now, it is eagle only. Sorry. ._.

There you go! A guide on understading eagles and the weird things they can and cannot do. If you have any suggestions, or things you do involving eagles, please put it in the comments below.
(Also if there are any typos. I checked, but I probably missed a few ._.)