Is it possible to have a different third-level domain name for a blog, e.g. "blog.website.com" instead of "website.com/blog"?

Site URL: http://www.drdawnscoachingplace.com

I would like to add a blog to my website, and ideally the blog would have the URL "blog.drdawnscoachingplace.com".  For whatever reason I feel this looks more polished and professional than "www.drdawnscoachingplace.com/blog".   

Does Squarespace support this?

You cannot use a different subdomain for a blog page on the same site. However, this shouldn't be an issue because URLs are becoming less and less important. 

That certainly is so.