Help With Solid Color Layering

I think a major part of the problem with this image so far is that, as far as Photoshop see it, “it’s a trap!” You’ve been looking for a single blending mode that will change both the black and white text to blue, but the problem is that blending modes often use tonal values to determine how much to change something. Because the text is at opposite ends of the tonal scale, it may be difficult to put both the black and the white text on the same side of a single blending mode result.

The approach I did in the demo below was to treat the white stripes separately from the red stripes. (This is not the only correct answer.):

1. Make all text 50% gray. I like to do this because it makes it easier to change color by area, using other layers and blending modes.

2. Draw a shape layer with a rectangle, apply a fill color of blue, and the Color blending mode.

3. Draw a shape layer with multiple flag stripes as rectangles, apply a fill color of red, and the Color blending mode. 

4. Draw a shape layer with the alternate stripes as rectangles, apply a fill color of white, and the Color Dodge blending mode. (There are other blending modes that work just as well.)

Do you want to place a blue solid color layer in the top left corner, beneath the text? Your layer should be below the text but above the layer with solid black color. Please post a screenshot with the layers panel expanded and visible.