If the government were doing it, they would be doing it already, and you wouldnt need to install a patch or anything, you would be completely unaware. So don't worry about that.
I don't think that logitec apps should prompt you to update, I know my mouse doesn't. Instead of using this suspect icon, you can go directly to the logitech website here:
http://www.logitech.com/
and use the support center at the bottom right of that page to download updates for any hardware you have. That should shut it up. If it doesnt then it's probably malware or something and you can follow curly's steps to cleaning up your PC that are stickied somewhere.
Someone else might know of a logitech product that might have some kind of update notification.