Has anyone experienced ergonomic problems with the Magic Mouse?
You're seeing some first indications of RSI because you're using your hand in ways you're not used to, and possibly because the mouse doesn't fit your hand. 

Switching to a normal mouse is a good first aid, especially since you noticed that the problem disappeared or became less of a problem. For some people, it just takes time to get used to new movements, while others won't be able to use the new device for longer periods of time. Those unlucky people should not force themselves to use devices that aren't comfortable for them, and this might include you.

The problem with the magic mouse, IMHO, is that it's too small to really be useful as a gesture device because you're gesturing with the same hand that you use to hold it. That's bound to be awkward, which stresses your hand's anatomy, and some people develop problems from that. Others might not notice any problems -- everybody is different.

Solution:


Change to a different mouse.
If you must use gesture input, use a separate device for that. Apple offers the magic touchpad, which is a simpler and smaller version of the original FingerWorks iGesture series of devices which you can still buy used on eBay. And their laptops have multitouch trackpads too.