Efficient workflow for creating identical animations at 16:9 and 1:1

First option: Drop your HD comp in the Square one and collapse transformations and/or scale and position the nested comp. 

Second option: Do as Mylenium suggested and make a master comp that is bigger than both with the appropriate guide layers. A shape layer set as a guide layer with a 1920 X 1080 rectangle with a stroke and another 1200 X 1200 rectangle would work just fine. Just put the guide layer on layer 1, then shy it to hide it from the rest of the comp. You might start by just resizing your main comp to 1920 X 1200. This is how I would have started the project:

If you are editing a comp longer than a couple of shots or trying to make a movie, use Premiere Pro. It has built-in tools to intelligently resize sequences for social media. In my experience, the AI (artificial intelligence) in Premiere Progets it right about 90% of the time. 

Well, typically you design such stuff in oversize comps with safe areas mapped out and visual aids/ guides and then it would be as simple as nesting and scaling comps, give or take some very specific tweaks. In your case you probably will have to do it a different way. Instead of trying to squeeze the larger comp into the smaller one, duplicate it and convert it into a 1920 x 1920 one, then modify the elements that would expand in the extra space at the top and bottom. Then only nest it for final rendering or scale it in the render settings.

Mylenium