Cooking eggs on cast iron
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/