I’m having this same problem and I’m in contact with Squarespace support as I believe this issue is something on their end. I’ll let you know how I get on.
I was going through the same routine of importing posts a few weeks ago (probably around the time you posted this, Steve) and things were fine. This was on a local machine. Then, yesterday I signed up for eHost and tried the import only to have all the images fail… well, almost all. Five images imported, and these five images are the only ones that will import each time I try the import either locally (where it now also fails) or on eHost.
I ran through the XML file generated by squarespace and selected 8 or 9 random URLS for attachments, loaded them into the browser and they appeared just fine.
So some things I’m pretty certain of:
1. The media is available. Browsing their URLs shows this.
2. It’s not a firewall or directory permissions issue as those five images are imported without fail, and I can upload/install images/plugins/theme just fine.
3. It’s not malformed XML. I tried with an older file (from October) as well as a recent one.
4. It’s not an environment/server issue. I am using the same version of Apache, mySQL, PHP, and I have tried this with WordPress 4.3 (which I was using when it worked) and 4.4.
5. The wordpress folks should just make a squarespace importer ??