Yes, your “Flag Icon.psd” file can be used as the background for The Expeditionary icons, however, those icons have additional graphics on top. For example, the " British Expeditionary Company" has a rifle icon on top of the flag background (per screenshot below). Therefore, your “Flag Icon.psd” file cannot be used to replace The Expeditionary icons, unless the modder uses software to corp out the icon on top and place it on the “Flag Icon.psd” file
In terms of inspiration flag, your “Object Flag.psd” file can be used to modify the inspiration flag model in the game. I tested the American one, replacing the “spc_americans.ddt” file after transforming the “Object Flag.psd” file works perfectly fine. (For Mexican, I assume you could replace the “mexican.ddt” file instead, but I have not tested it out)
Unless you were talking about inspiration flag icon, then yes, it would need a new file to replace that icon