Without actually having the digital assets to look at, it is a real cr*pshoot to guess as to what is going on here.
What is the pseudo-American flag in term of object types? Vector, raster, what? Perhaps you can point us to the objects to examine?
The elements are Vector, and when printed from Illustrator are perfectly fine. If I am to rasterize them before printing they also print perfectly fine? I could upload the flag elements for you to examine if that would be helpful?