When Army Sgt. Patrick Hart decided a decade ago that he would not serve in the war in Iraq, he expected to follow the same path as thousands of American war resisters during the Vietnam era and take refuge across the border. But after five years of wrangling with the Canadian immigration system, he came back to the U.S.—and ended up in a military prison. The country that once welcomed war resisters has developed a much different reputation during the conflicts in Iraq and Afghanistan: Supporters say no U.S. soldier who has sought legal residence in Canada, either as a...