So, I've played through the tutorial and have tried to start a new game but after I click "attack" it starts the into scene for a bit then the program just shuts down and I'm back at the desktop. Any ideas on what's going on?

Having the same problem at the exact same time as you. Windows 7 64 bit, mine the program does not quit though it just minimizes to the taskbar and refuses to come back up and I'm forced to use task manager to end the process.
Thought maybe it was my 1920 x 1080 resolution but tried lowering it and still nothing.

Going to keep trying to sort this out and if I figure out a solution I'll be sure to post.

Well that did not take long, got it working. Not sure exactly what I did as I did something then changed it back and tried the game and it worked.
Not sure which of these steps fixed it but it is working now.
I was trying to run the game from Galaxy so thought I'd go to the games folder (star wars battle front II, Gamedata folder) and try some compatibility modes on the .exe itself.
Once there I right clicked on the batllefrontII.exe and choose properties, then went to compatibility tab and choose to run in compatibility mode for XP SP3. Clicked apply then thought better of it and thought I might as well just try running the game from the BattleFrontII.exe file itself first so took off the compatibility mode setting that I did and just tried the game from the BattleFrontII.exe and it got past the part it was crashing (right after the cone head guy is fighting and the tank blows up) in the cut scene and now seems to be working, I have not gotten into game yet but have gotten to the troop selection screen.

Not sure what part of that did it but something did, game seems to be working from the BattleFrontII.exe or Galaxy as well.

EDIT: That worked for one mission so seems maybe you need to run the game in compatibility mode I'm running Windows 7 and set it to windows 7 compatibility mode and it seems to be working again. No clue what is going on but it is real finicky.

Well back to square one as the minimize to desktop and not being able to bring back up happens after every mission. Running in compatibility mode does not seem to help.

UPDATE: This game now started working, don't know if GOG fixed it with a patch or what happened. The only thing I really changed on my computer is I updated my sound/audio drivers to the new ones for windows 10 even though I do not have win 10 yet.
Actually have gotten through 2 missions so far without a crash, whereas I could only do the first mission before and since I never got to a save point just had to do that first mission over and over for the single player campaign.

So OP if you are still wanting to play this game and are monitoring your forum replies give the game a try now.