In summary I have a java program that writes a file and I would like to now take that file and upload it to my Squarespace website. From here I am trying to figure out what my next steps are.
Unfortunately, Squarespace currently only offers APIs related to commerce (inventory, orders, products, transactions) and forms. There is no file storage-related API.
The only ways to upload files are via the WYSIWYG editor (via the add-link button), via the CSS Editor (for image assets) and via developer mode.
Squarespace 7.0 sites (current sites are 7.1 by default) that have Developer Mode enabled do support Git and SFTP. Theoretically, one could develop a system that utilized one of those methods in order to add files to the site (which would then be accessible via /assets), but that is only theoretical and far from ideal.
I'm afraid there isn't a good method for automatically uploading files to a Squarespace site.