Squarespace has the built-in Aviary photo editor, but it's currently not possible to make a SQUARE CANVAS, while keeping the "original non-square format" from distorting. So, the only good way to make the images square, would be to edit the images offline.
Most Squarespace processes (like in "Summary Blocks") will auto-crop the images into selected ratios, but at the time of this posting, blog/page thumbnails will just use the original image ratio.
In the world of Instagram, where everything is square, there are likely many apps/sites that can quickly process your files into square/1:1 ratio images. Here's a site from a quick search for "make square images" http://www.orbling.org/squareit/ (And if your images are on your iPhone, you can use the "Whitagram" app to "crop your canvas")
Of course, you would be better-off "batching" all of your images before uploading them, which can be done on Photoshop, Fireworks, etc. —also see the following Smashing Magazine post on the topic: http://www.smashingmagazine.com/2008/10/02/15-useful-batch-image-processors/