Actually out of the draftees about 15000 gets to do military service in sweden (last figure I've read about), and if you want to go career you're in for alot of competition.
there are tens of thousands of people applying for the job and most spots are allready filled. Good for the military (they can pick the most promising ones) bad for the individuals (less chance of getting in).
You can check the swedish site soldf.com's forum for a similar thread like this one, if I remember correctly there should be a thread already with exact info on what you need to do and have, and yes greencard is required, and for any security-clearance you have to be a resident. I was thinking about doing this to a while back and would just like to say that you should think things through first, if you try for this you can expect to wait a long time just for a green card. I'd advise you to call the US Embassy in stockholm and have them give you the phone-number for the recruiter in germany.