How do I defend myself against the eagles?

There is no way to defend yourself against eagles once they attack you.

You can try to spot them before they swoop down and attack you, but you're going to look like a crazy person looking up every 5 seconds.

There's no way to shake or melee them, but shooting them before (or after if needed) is a good approach and isn't that hard.

1 - The Warning Cry

While it's true that eagles occasionally give a single, long call, before attacking they give a very distinctive cry: three high-pitched "skrees" in quick succession. Sometimes these are followed by the standard call.

The next time an eagle attacks you, just stand there for a while and let it come back for successive attacks, healing as needed, and learn to recognize the attack cry.

Over time that cry will probably start giving you an unpleasant jolt even if you're not specifically listening for it. (Thank you, Dr. Pavlov.)

2 - Initial Defense

When you hear the attack cry, either head for cover or look up quickly to counter-attack. From the start of the first "skree" you'll have 3-5 seconds until that unpleasant bird-bite cut scene.

With luck that'll give you time to look up and switch to an appropriate weapon. Shotguns or guns with a high fire rate (machine, SMG) are ideal.

3 - Payback

If you don't kill it the first time around, quickly switch to your camera and tag it, then wait. If you stay in the same area it should attack again fairly soon, and this time you'll be ready to eliminate the threat.