Having seen the Ameriware cookware at a local store, I did a search and happened upon a "Cooking for Engineers" site that revealed the non-stick coating to be a product called "Eclipse" (made by Whitford), modified by the addition of titanium. Eclipse does contain PTFE, polytetrafluoroethylene (Teflon is DuPont's brand of PTFE coating).
The modified Eclipse coating appears to be very durable. From what I've read, the concerns about PTFE relate to the use of coated pans when the integrity of the coating has been compromised (scratched or chipped) or superheated (a dry pot forgotten on a burner). That said, I tossed everything with a non-stick coating in my house because I'd rather err on the side of being safe and tend to be untrusting.
I've found that if I preheat a good, heavy-bottomed stainless steel or anodized aluminum pan, then apply some shortening (usually, olive oil) and allow it to heat through before adding food, I do O.K. And, as I'm sure you know from your own reference to owning a cast iron skillet, a well-seasoned one works well. Maybe getting a second cast iron skillet would be a good alternative, in a different size/shape to make clear which is for eggs and which is not.