I think it was just a corrupt download, if you delete it and download again, it should get fixed, at least regarding the download part, but I'm not sure about the rest, flash actually working correctly with sound everywhere and all.
I don't think it was really a matter that a clean install would fix, unless you previously had a heavily modified system, with packages from different distributions and whatnot.
Anyway, did you try the non-free flash from the repositories?
In order to do that, add "non-free" in your /etc/apt/sources.list file. The line should be something like this:
deb http://ftp.br.debian.org/debian/ stable main contrib non-free
Except for the "br" part, change it for whatever country "code" you're in. Save it, then go to synaptic (I always prefer that as I never know the package names to use via command line), reload, and set to install.
If that does not work, but you still want to try other package options to see if that alone would fix it, then there's the least recommended option, adding the unofficial sources from here:
(the lines to add to sources.list are there)
and install the same packages I have, and I don't even know if it is possible on squeeze. But I guess the nonfree should work. I'm currently using packages from there (adobe-flash-player and adobe-flash-player-browserplugin) because I've had a few problems with youtube videos some time ago (just weird colors, like the red, green and blue being displaced, looking like some crazy image for 3d glasses), but I'm not even sure they're any different from the standard non-free version, the problem I was having is gone, but I'm not sure about what really solved it.
And, anyway, it could still be that installing from this file you tried to download would work as well, try whatever you feel more comfortable with first.