Other people may not think so, but I can't imagine mixing coconut oil with eggs. ICK. I like butter. Use ghee or clarified butter if you can because it doesn't burn.
The pan has to be HOT. Hot enough to sizzle the eggs when they hit the pan, not just warm enough to melt the butter. This is important. The underside of the egg liquid has to cook and solidify immediately. If the egg stays liquid for even a few seconds, then the melted butter underneath will float to the top. The eggs will touch the pan without a layer of fat and they WILL stick.
5'0" female, 45 years old. Started Primal October 31, 2011, at a skinny fat 111.5 lbs. Low weight: 99.5 lb on a fast. Gained back to 115(!) on SAD chocolate, potato chips, and stress. Currently 111.