OMAHA BEACH, France—Jim Kosinski sensed something was wrong when he noticed the American flag wasn't flying here on this windswept shoreline, the final resting ground of thousands of U.S. troops who died during World War II's D-Day invasion.

Moments later, the 71-year-old military veteran—who traveled from Ventura, Calif., with his wife to pay homage to the fallen soldiers—saw the sign affixed to the cemetery gate: "Due to the U.S. Government shutdown this site is closed to the public.

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