Compress makes the file smaller. It doesn't change the image's shape or its format.
Square makes the image a consistent 1:1 shape, either by adding padding around it or by cropping. It never stretches or upscales a small image.
You can do either or both. "Compress and square" does both in one run. Compression has three modes in Settings — Conservative, Balanced (the default) and Custom — and you can preview the result on a sample image before applying.
Not sure which mode suits your images? Chat with us and we'll help you pick.