So, I tried the update again today and had the same result, it wouldn't proceed with the install. Just gave me a message saying that the install failed. The amber LED was still blinking.
So here is what I did to make the install work:
I didn't attach an Ethernet cable to the MBP. Just left it hooked up to my DSL modem.
I opened the AirPort Utility and selected "edit". I saw a flag in the "Back to My Mac" box in the "Base Station" screen. I selected "edit" and I entered my apple id and password. Then I selected Update at the bottom of that screen and it updated the Back to My Mac data. It took quite a long time.
When it was finished, the Amber LED stopped blinking and was on steady.
I closed the menu and then clicked on the AirPort Extreme icon in the AirPort Utility. It opened a screen with the setup detail and I saw that it still had version 7.7.2 of the firmware installed and a note that there was an update available. I told it to install the update and it installed, taking quite a long time. It was hard to wait as I was concerned that it was not going to do it right. If a firmware update doesn't go right, it can really mess things up. It finally finished.
I checked and version 7.7.3 was now installed, but the amber LED was blinking again. So, I selected Edit in the AirPort Utility and sure enough the Back to My Mac id and password was missing again (as edbedb suggested above). So looks like the install does wipe out the apple id and password for Back to My Mac in the AirPort Extreme. So I again entered my apple id and password. Let it update again and the blinking light went to steady. Mine is now updated to 7.7.3 and the amber led is on steady and everything is working.
So, if anyone is still having a problem with this update. Follow the above directions and it will install.
Wonder why Apple doesn't give us some hint or help in installing this update?? There is no help in the AirPort Extreme help file. I wasted a lot time searching for some instructions on updating the firmware in the help file system. Nothing there.
I'm surprised that Apple doesn't monitor these forums, since they are on the Apple web site. Thanks to those who do monitor here and offer some good suggestions. I'm impressed with those who are willing to help!