I like to use flannel for my rag quilts. I've made three of them in flannel.

I’ve seen pictures of some really nice red, white & blue ragged quilts. I’m sure yours will be great

if you still want to use the 3 layers of cotton fabric, would they fray up better if cut on the bias to begin with?

I love cotton flannel, just cotton isn't soft enough to cuddle up with. Have made a few for boys, denim squares and flannel, sturdy and yet soft.