I think I found a fix
Like many folks on this forum and several others I found while scouring the internet for a solution to this problem, the HX550 worked flawlessly on my old 4S. Many posts started out by saying "this worked on my 4S and it doesn't work with my new 5". When I got my new iPhone 5 at the AT&T store, I jumped in the car, paired my HX550 and drove home. The HX550 did the "click" noise and hung up on several calls (wifi was turned off, so this didn't seem to be the problem as others above have mentioned). When I got home, I plugged into iTunes, upgraded iOS to the latest version 6.1.3. HX550 still didn't work. I tried to "forget this device", reboot, and re-pair. Still didn't work.
I wondered if this had to do with the order in which I paired my new iPhone 5 and upgraded the OS. On my 4S, I originally used a different bluetooth earpiece. I upgraded iOS as I went along. When the old bluetooth earpiece battery died and wouldn't hold a charge, I bought the HX550. With the iPhone 5, I paired it when the iOS version was still whatever it was when it left the factory and then updated iOS when I got home.
Today, I used iTunes to reset my iPhone 5 to the factory settings. Then I updated iOS. I loaded all my apps and synced my music, photos, etc. THEN, I paired the HX550. It seems to work now. I have not lost bluetooth connectivity even on long phone calls longer than 10 minutes (wifi is turned on and I've used Siri). Granted, I haven't conducted extensive testing over a period of several days yet. I'll report back in a few days with more testing/usage results.
Good luck to everyone!