First I would remove the module and check the db it has been removed properly Admin > Modules > Payment > Square... then remove the files and make sure they are not on the server... then reinstall everything.

Test the install.

Get errors again...
I would have to look at what is going wrong to see if there was a non generic tracer in the error log perhaps. Or force an error to see if there is an old reference on the server. Next I would try an older API version just in case the 'newest' one had a bug. And then start checking if the server version and PHP version mismatch along with the older PHP bridge these servers use is faulty.

You could post the error log get more eyes on it or hire that Twitchtoo guy to have a look - coffee is always a good place to start!