As you say, some actual US military first aid kits have a red cross on them...here's one I've seen in real life: https://cdn.shopify.com/s/files/1/0106/5435/1460/products/1a...
And, FWIW, those first aid kits in big green plastic boxes are mostly used by regular soldiers. The medics have different kits in backpack looking containers called an M-9.