I took the liberty of testing this as another customer brought this to my attention. It seems it's not specifically an issue with Flash or Firefox but rather a Java + UAC issue. I'm making a few assumptions here as everything started working for me when I did the specific "Java 1.6 Tweaks" (also noted in the "How to ThinApp Newer Versions of Java" blog post just prior to step 4).
Specifically, the added Java browser helper objects seem to be causing the UAC issues. Not sure why it only appears after launching Flash first though.
Anyways, I captured the above versions together in a single package on a clean XP SP3 system and after the java tweaks and Flash MMS.CFG settings, was able to get the entire thing working on XP, Win 7 32-bit, and Win 7 64-bit. Even did the "Dirty, Washed, Clean, Production" tests as well and all worked.