I recently looked to implement the proper markup on my blog, hosted by Squarespace, to provide Twitter Cards the information it needs to render a preview of my blog posts.
Why this isn't either on by default, or an available option through the GUI, I'm not sure.
UPDATE, Dec 16 2014: it seems Squarespace 7 does not play nice with Twitter cards. Squarespace sites that previously had Twitter cards (like this one) have lost their cards and the validator is unable to see any meta data. I reached out to Squarespace about the issue; they said their developers have this flagged as a known issue and are working on a fix as soon as possible. Here's hoping to a quick resolution from here on out. Follow me on Twitter where I'll be sure to announce the fix as soon as I get word from Squarespace. Thanks for reading!