The latest version of SquareDesk (0.9.5) is now available for Mac OS X, and has been tested to run on Catalina. You can download it here. Automatic downloading from Dropbox is still available, for those who prefer that.
The README.txt file has information on what’s new, what’s changed, and what’s been fixed. A list of open issues and feature requests, and the source code, can be found over at Github.