Do you mean that the assets will reorganize within the square frame to have the same look and feel as they were in the 16:9 frame? Or as Tero suggests you just need a crop?
For just a crop you can drag your16:9 into another comp that is 1080×1080 and that will automatically trim that first comp live – so you can move things in the HD comp and see how it looks in square too without remaking either.
The other way for the same look and feel idea, is to position your assets in relation to the center and frame edge, not simply by scale and position (usually rotation looks after itself). Its best to use expressions for this so they can calculate things like even spacing and distance from edges. Then you can use a second part of those same expressions to alter the assets for your other aspect. When the second half of each expression is made active these will move assets to the same kind of relationship they had in the HD comp but formatted for a square, and then render out of a parent square comp that is cropping the original HD. You do have to put a switch on the expressions so that you can turn sections of them all on or off at once. Use the check box effect and link the expression math to that one effect. Remember that expressions feed the last number calculated to the attribute, so if let the code continue the first chunk for the HD version will be ignored. Basically it can be as simple as a giant if/else with the second calculation inside the “else” braces.
This can be easy or hard depending on the content design. But if its a pipeline for multiple screens of different formats that will be used over and over, its great because you just create for HD and let AE do the reorganization for the other formats.
When we do this, we also include sliders out in the square comp that link to the original expressions on the inner HD comp, so that from the square comp we can tweak positions or scales without affecting the original infrastructure.