Yes, it is definitely a Star Trek reference. In the original 60’s series, the shirts were either gold, blue or red.
The command staff wore gold and the science/medical staff wore blue. When they needed actors to play regular crew members, presumably enlisted personnel rather than officers, they were typically dressed in red shirts. But both Lt. Uhura and Chief Engineer Scott both wore red, so it wasn’t always consistent.
Away missions always included a few extras in red shirts. Obviously, they couldn’t kill off the stars, so if anyone dies, they usually wore one of the red shirts.
Has anyone heard how Aaron is doing medically?