I’m interested in the general problem you’re trying to solve, but can’t promise too much yet.
There are two common approaches for logging into Discourse from another application. The first is to use the other application as either an OAuth2 or OpenID Connect identity provider. The second is to use Discourse’s implementation of SSO (DiscourseConnect.)
As far as I can tell, Squarespace sites cannot be used as OAuth2 or OpenID Connect identity providers. That rules out the first approach. If I am wrong about this, hopefully someone will let me know.
Squarespace also doesn’t currently support DiscourseConnect, but it might be possible to develop a Squarespace Extension that supported DiscourseConnect and possibly other Squarespace/Discourse integrations. Developing this would be somewhat complex, as the code that handled Discourse logins would have to exist on its own server. I think it would have to be a paid extension.
I’m not certain that this will be technically possible, or if there would be enough demand for it to justify the development time. I’m curious enough to look into it some more though. I’ll update this topic with what I find sometime next week.
One thing to note is that if you’re not yet committed to Squarespace, the Discourse WordPress plugin supports DiscourseConnect. It is very straightforward to configure.