Many actually think the reverse is true, with the flu originating in the US, and troops carrying it to Europe. The first known outbreak was in rural Kansas...followed closely by a large outbreak at Camp Funston (Fort Riley, KS), and from there it went around the world.
Around 10 years ago, I designed a military vehicle parking facility on Funston. I was out there when the contractor first turned over the dirt...and suddenly wondered if the virus may somehow still be alive beneath the surface....apparently not.
Your premise is still correct though - soldiers move around a lot, and live in close quarters with poor hygiene. A perfect breeding ground for spreading disease.
Link to info about origin:
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC340389/
You know, I think I remember reading about the Kansas connection somewhere. Something about them burning a large number of horse carcasses (weird correlation, but at the time I think the locals though that was the cause)?
You were not far off. Anthrax sleeps in the soil until conditions become just right. The grazing animals drop like an ocean tide.