I think the fabric in question is fine.
No matter what we do there is a chance someone will not appreciate something. As the daughter of a Vet (sister, niece, etc., etc.) I don't think there is a problem with flag images on fabric. Or the use of stars and stripes, either as a print or a design element. All such things have been on novelty fabrics since at least the civil war. While I myself wouldn't use one, I wouldn't think most people would have no problems with a flag applique/embroidery like you'd have on a shirt or a pin either.
But no actually cutting up a flag or using one for a back, even a print 100% cotton one.
And there are some veterans who aren't necessarily feeling all rah-rah red, white and blue but that's a different issue.