For a while, I was actually able to upload the Flash games I've made to the Squarespace 6 servers and then link to them in a custom "code" block - but unfortunately, policies change, and all linked material started being served by Squarespace 6 with a content-disposition set to "attachment" - an attribute which apparently causes recent version of the Flash Player to ignore the Flash content! I submitted various forum postings and help requests on this issue, and received nothing remotely helpful. I've therefore decided to use Amazon Web Services' "S3" service, which lets you host simple web files up to a certain limit for free (and then afterwards you pay only according to the used bandwidth). This also has the benefit of not forcing me to encode my Flash games as a single swf file, which in some cases would be gigantic.
The general lesson, I think, is that if you are considering Squarespace 6 and are looking to do anything remotely sophisticated, give it careful thought. The staff won't support you, and there may very well be problems with no solution. It's a shame, given how much I enjoy the look of the templates and the simplicity of the content management system.