I/you will get an "i'll see if i can tinker and see if i can get the video player app running on cm7 - i'll peruse the kindle fire section to see if they are having the same problem.
edit: checked over at XDA-kindle fire and they got nothing-ie cm7/9 does not run the video app either from what i can find.
I then look over at Amazon.com's listing for my Devices...i have 6 devices listed as my Nook Tablet:
a few listed as "Barnes&Noble BNTV250", which are the ROOTED version of the Nook tablet and CAN run the video app with no problems.
a ferw Listed as "bn Nook Tablet" is the Tablet using CM7(2 tablets listed one was the internal version one is the microSD external version) and neither version can run the video app, one thing i will try initially is to change what the device lists as and try to get the Tablet device to ID itself as Barnes&Noble BNTV250...
Spoofing the device ID will get you nowhere... Both Amazon Video and Google Video do not work on CM7 because they both detect that you are rooted. I haven't figured out exactly how (honestly I haven't been working on this) but it has to do with files found in /xbin or with even having an /xbin directory. I found this out when neither would work on my rooted stock NT with Busybox installed (creates and installs into /xbin). By renaming /xbin to anything else (or not installing Busybox) and unrooting with OTA Tempunroot then rebooting you can get either video app to work. CM installs system files to /xbin rather than /bin so you can't just rename /xbin (your sytem will not run properly and you still won't be able to get the video apps working).
So that's where this currently hangs: either find out what file(s) in /xbin can be renamed or removed to appear unrooted, or figure out how to install CM components to /bin instead of /xbin.
Hope that helps, please post anything that does or doesn't work.