I'm not familiar with Squarespace. Is it a service (hosted) solution or do you run their code on your own server?
Do you have FTP or sFTP access to your site?
If it is a hosted solution and you cannot upload entire filetrees to your site then I do not think that you could use OST. Unless you setup a regular web hosting account somewhere else that you can do these things with . Then it would be a matter of setting up a hostname. (like support.yourdomain.tld) to redirect people to.
