The best setting to employ in this situation is back-button autofocus. It will put you in control of when the camera can search for focus and when to stop, and will separate the AF start/lock function from your shutter button, so that pressing the shutter button itself does not start up the AF hunt again when you least want it to.

You can set up the AE-L/AF-L button on the back of the camera to become the AF-start button by going to custom setting menu, and set F5 (Assign AE-L/AF-L button) to AF-On. Then, in the Autofocus menu, set A1 (AF-C priority selection) to release--this tells the camera to take a picture even if AF has not achieved lock.

Now, the AF hunting function is removed from the shutter. To start the AF search, press the AE-L/AF-L back button. To lock AF, simply release the button.

When shooting the eagles, press the back button until you see the focus is on the eagle you want and take the shot. Pressing the shutter button will NOT affect the autofocus (i.e., halfpress doesn't lock or start up the hunting again. With stationary subjects, you can use this to acquire focus, and then lock the focus (stop pressing the button) so you don't have to hold a half-press re-acquire AF lock repetitively.

You may also want to select only a single AF point, rather than using the entire array, if you think you can draw a bead on the specific bird you want.

In my experience when shooting raptors, 200mm isn't nearly long enough. I use a 400mm lens on a crop body and it's not long enough. Given that you're using a superzoom lens, you may also want to stop down for sharpness and added DoF, and increase your ISO setting (say above 400) to ensure your shutter speed is fast enough to eliminate camera shake blur from handholding as well as subject motion blur. Expect to crop and sharpen in post. Also, if you aren't already, consider shooting with both eyes open--it's easier with a telephoto or supertelephoto to narrow in on your subject this way.

Burst mode is a given, however, no burst speed in the world can make up for bad timing. It's best to go with short, controlled bursts, not spray'n'pray. And in burst mode, it may be more difficult for the camera to keep tracking in autofocus, so you may want to use CL instead of CH at times.