Be warned! This is a scam to get spyware and viruses installed into your computer. I have once clicked the button and immediately realized my mistake when I noticed that the browser is attempting to download an executable file that looked extremely suspicious. On researching further about the origins of the file, I have found that it comes from a spyware making company (of sorts).
If you want to check whether you have the latest version of Flash Player, just go to this page in the official Adobe Flash website. It will show you the latest version of flash player, and the version you have installed. Also note that Adobe Flash does not alert you of an update the way the scam does. Besides, videos will still play as long as you have a recent version of the player even if you don't have the latest.
I didn't get a chance to make a screenshot. When I do, I'll put up a screenshot here. The conditions where it happened to me are:
- Random videos from YouTube (shared by my friends in the Facebook wall)
- Facebook home page news feed
- Mozilla Firefox 4.0.1
- Reloading the page and playing the video again fixes the problem.