I'm running in W7 RC1, x64 with about 6gb RAM and an nVidia 250. For some reason, the default compatibility options prevent the game from loading past the initial black screen. If I turn off the compatibility options, the game loads up, but if I try to load any missions or training, the game crashes to the desktop after the planning screen. I don't get any errors. Oddly enough, I can do the skirmishes with no problems. Is anyone else having this problem? The game is supposed to work on Vista, and they're not much different.

On Windows 7 it says it stopped working when I try to load the game. It doesn't give any more information and I can't get into the game.

Try turning off the compatibility options, they are on by default, and I couldn't get the game to load at all with them on. Also, as with most problematic software, us the "Run as Administrator" option--even if you're currently the administrator.
It won't load any mission that has a briefing screen (you're limited to instant action only), but at least it works...
Question: are you using 64 bit? I have my suspicions that this game has severe stability issues on x64 systems.

Yeah I'm using 64bit.

Yeah, turn off the compatibility options. You will only be able to play instant action.
If you've got a copy of XP, you could use a virtual machine (not the one free from microsoft, it doesn't support 3-d acceleration) like VMWare server.