NOTE FROM FLASH-AID DEVELOPER: Flash-Aid security issues addressed here have been fixed in version 2.2.0. The extension now uses SSL for downloading beta plugin data, there is no more redirection and beta plugin is only installed if hash value is authenticated.
While Flash-Aid (should) work, I personally do not like the way it installs Flash system wide. It also downloads the flashplugin via an unsecured redirector on updates.webgapps.org so it is possible that somebody tricks you into downloading and installing a malicious version of Flash (or something completely else) by using a MITM attack on a public network.
The actual fix for the initial problem seems to be that 32 Bit Flash on 64 Bit systems sucks. So all you need is the preview version of the 64 Bit version of Flash codenamed "Square". You can get it here. The compressed tar file contains a single file,
libflashplayer.so. You have to put this file into the directory
~/.mozilla/plugins and restart Firefox.
I put a simple script to achieve this here. You can run it by calling
wget -O- https://github.com/mss/getflashplugin64/raw/10/getflashplugin64.sh | /bin/bash
Of course you still have to trust me not to do anything nasty, but at least you can be sure that nobody intercepts your download.
One thing you have to watch out for: The current Preview 3 is from November 2010 and carries the version number
10.3 d162. There was at least one security fix since then and the current version is
10.3 r181. I recommend to use Flashblock. You also have to keep this version of Flash up-to-date yourself.
P.S. to @lovinglinux: Feel free to reuse my script in Flash-Aid. Maybe you could add an option to install the plugin locally only. And you could request the update JSON file from your GitHub repo and get HTTPS for free. It could also return the required download URL which I parse from the HTML and you wouldn't need your redirector either.