The Magic Mouse and YouTube videos don't have anything to do with each other. I switch between my Magic Mouse and my track pad all day, and it has no effect on YouTube.
YouTube playback is browser and Flash Player related.
Check your Safari Security preferences to be sure that Plugins are allowed.
Worst case, do a "clean Install" of Flash Player. Instructions (with illustrations) are here: http://forums.adobe.com/thread/1343084