I've seen this question come-up on the Squarespace forums in the past— the need exists where people want to have a unique login template for their site.
Well, it's cake. It just takes a little juggling knowledge.
404 Error Pages
This same technique applies for your Squarespace 404 Error page (the page a visitor sees when a "page isn't found" because of a broken link). The difference is— the Login & Error pages have no "Page ID", so it's not really that easy to add a special design for 404 error pages. (like these)