It's your country, too.

Wear that flag with pride, just as Ginsue said. You're not being a poser, and you're not indulging in stolen valor. I've known people that wear unit patches of loved ones that were KIA, or relatives that served during a war. They're eager to tell people about their friend or relative; they don't claim that they served.
My Dad didn't serve- he had high blood pressure and flat feet and they would not let him in. For every military member, there's a network of support in the community. That's you.