Squarespace's propaganda goes on about "Total Data Portability", and how "your data is yours to keep". However, when it comes to moving to another system, there's no automated method to get your images and other files out. You can use a "data export" to create a Movable Type formatted export of your posts (which will import fine to most blogging engines), leaving your images in their proprietary file system, to which you only have web access, and no functionality to create an archive for downloading.
I asked Squarespace support if there was any other way to get our images out, and their response was "Currently there is no feature for this -- each file will need to be downloaded individually from File Storage, sorry about this."
I asked if we could get a Squarespace dev to compress the directory so we could download it, and got various permutations of "that feature doesn't exist, but I can suggest it as a future feature" from 3 different Squarespace support people:
Fortunately, while all this frustration was going on, I figured out a sneaky little system to allow me to download all of the images directly from their web interface. It uses Firefox and the DownThemAll extension.
How to rescue your images from Squarespace
If you don't already use Firefox, download and install it, and make sure you check the "Use Firefox As Your Default Browser" box. Welcome to Modern Browser Land. You're going to like it here.
Next install DownThemAll, and restart Firefox.
Login to your Squarespace account, and go to the "File Storage" area. Hopefully, like Dubspot, you haven't been too organized, and most of your images are in a single directory, like this:
Right click on the page, and select "DownThemAll".
We only want the posted files, which are held under a directory called "storage", so click the URL column header to sort all the files by URL. DownThemAll had skipped out some files - PDFs and SWF mostly.
Click "Start", and you get all of your files back. Well done.
Unfortunately your journey isn't over yet. Squarespace's Movable Type formatted export includes a lot of cruft in the HTML, which takes a bit more work to fix up. Details to come.