Have you tried using the manual installer? For details on that method see http://kb2.adobe.com/cps/191/tn_19166.html#main_ManualInstaller

I find the manual installers better than using Adobe's download manager to install Flash.

I have the same problem trying to download the manual installer. I also tried it using the Admin account thinking that some security or user problem was in the way.

See also http://kb.mozillazine.org/Unable_to_save_or_download_files

It is possible that your anti-virus software is corrupting the downloaded files or otherwise interfering with downloading files by Firefox.
You can try to disable the real-time (live) scanning of files in your anti-virus software temporarily to see if that makes downloading work. Don't forget to re-enable once you are done downloading and installing Flash.