I noticed in the mod that some units have the appropriate flag for their nation within their unit texture. How does this work? I might try it for a mod I'm making.

I have to admit that I do not know. I haven't made any unit models so I don't know how they implement the flag graphics to those units. Maybe try asking the same question on unit graphics forum section?

Yeah, I'll try that. Thought I might ask here just because this is the only mod I've seen that has kind of graphic trick

Most likely the flag is included in the unit texture, if you can give unit names where you've noticed this, I can check the textures from them. All I know is that team (civ) color can be set to unit model somehow.