Kosher salt and a small amount of vegetable oil. Scrub the pan with the salt on a rag or paper towel, if there are stubborn bits mix a couple drops of oil with the salt, wipe dry with clean towel. If you use a wet method to clean the pan re-heat it after cleaning to make sure it is completely dry before storing.
All my attempts to "clean and repair" a sticky-seasoning layer have ended badly as well. I know who already snickers in the background, but my opinion is: start clean. Give the pan a lye bath and a good scrub, then reseason with a better seasoning.
"The new hotness" of flaxseed oil goes back to the fact that it doesn't go rancid easily, without heat it just dries out making a film which can protect things like wood. I bought into it, and seasoned a pan with it, and wondered why all my seasoning tries were unsucessful (I stripped and reseasoned 3 times, I think). I came to blame it on the smooth forged iron.
Then I got Cookwise and read the chapter on fats. And surprise, what does it say? That PUFA stick to everything around. (Makes sense, if you stop to think about it - those unsaturated bonds are unstable, they itch to break apart). So, when I got my next iron pan, I seasoned it with three (very thin) layers of flaxseed oil (for a solid base which will stick to the pan and the finishing layer) and then two layers of lard. After the oven, I heated it from brown to almost-black on the stove (empty).
This seasoning turned out perfect. I made crepes on it, re-oiling once every 7-8 crepes only (no fat in the batter). The pan released them like a charm. The old flaxseed-seasoned pan still sits around with some carbonized matter burnt onto the seasoning, and waits for a lye-flaxseed-lard session.
Of course, I would still take some caution with new seasoning and oil well the first few applications (I only tried the crepes after I noticed that less problematic items work great). And if you tend to often fry with the fashinably-healthy nonsaturated vegetable oils, don't heat them too much, else they could bake in a sticky-seasoning layer onto the pan. In the worst case, if you do get sticky-oil (but not carbon) buildup, try adding a new lard layer before you strip-and-reseason.
Best Answer
You hit the nail on the head. The seasoning is coming off. It's just chunks of fat/oil with bits of black rust, or black oxide.
The solution is to either stop using the pan to cook scrambled eggs, or strip the pan down to the cast iron and re-season it. Seasoning is a two part process. The first part is developing the layer of black rust, and the second part involves coating the pan with oil and then using heat to turn the oil into a polymer that provides a non-stick surface. If seasoned properly, you really shouldn't see any issues.
There is an interesting series here on the subject of seasoning cast iron:
http://sherylcanter.com/wordpress/2010/02/black-rust-and-cast-iron-seasoning/ http://sherylcanter.com/wordpress/2010/01/a-science-based-technique-for-seasoning-cast-iron/