I was using my D300 with a 300mmf2.8 VRII and a 2.0III teleconverter, this weekend. I was mainly shooting a flycatcher on a fence post. I was getting some nice handheld shots. A Northern Harrier was flying across the field and I switched to the hawk. Unfortunately, I could not get a good focus on the bird. I took many shots and All were way out of focus or very soft. Does the teleconverter affect the focus or was I doing something wrong? I am new to birds in flight. I will attach a picture. Thanks.
You will be hard pressed to get great images at 600mm handheld without tons of practice and good technique. My experience says that once focal length goes past 400mm the amount of sharp handheld keepers goes down dramatically. Also, the 20TC is agonizingly slow to focus, it hunts badly when focus is lost-I would suggest using a 14TC as that barely affects focus speed. Keep shutter speed to 1/1000 at the minimum and try to shoot at f5.6 to f8 for greater depth of field and focus accuracy. To help maintain shutter speed don't be afraid to crank the ISO up to 500, 640 or even 800. I see you were at 1/2500 and f7.1 so those settings were right on.
Shooting birds in flight takes practice but it's very rewarding when you nail it! Many photographers use only the central focal point, use af-c (continuous focus) and try to keep that focus point on the bird's head or wing butt so that the eyes are sharp. You must also learn to pan and fire with the bird.
Lastly, learn about the bird's habits and get closer by concealing yourself in natural blinds or get a commercially made blind. Birds see in full color and some raptors can see prey 8 miles away! Do an internet search on bird in flight photography and you will find a plethora of info. Practicing on semi-tame birds like pigeons and seagulls will strengthen technique.
If you are just starting out shooting birds you may want to take a day class from expert birder Andy Nguyen. He is also a Floridian and lives in the Orlando area. Check him out at wildwingsphotography.com and click on workshops.