Individual product pages don't seem to be indexed by Google. I can do a site:mysite.com search and they don't appear.

This may not be due to Square. Try searching on Bing or Duck Duck Go. Google is going through a weird transition right now and a lot of sites aren't being indexed. My own site, insurgent.ca, has been live for months and nothing I do in the search console seems to be able to get it indexed. Meanwhile, every other search engine seems to have no problem indexing my site at all.