Sounds like Google indexed mysitename.squarespace.com while you were creating your site. If mapped correctly, it will only index mysitename.com from this point forward.
If you're using a CNAME to map your domain, then the duplicate results you're seeing in Google will eventually be mysitename.com only. It believe it takes a bit for Google to drop the old indexing of your "squarespace.com" domain.
If you're using forwarding, that's a big problem for SEO. You'll want to use these instructions instead.
Hope this helps!
I'm concerned that my site may not be SEO optimized. I'm particularly concerned about the following issue:
I seem to have 3 domain names. www.mysitename.squarespace.com, www.mysitename.com, and http://mysitename.com. I only registered www.mysitename.com with godaddy. If I type in www.mysitename.com it goes to my site but it reads http://mysitename.com in the address bar.
In my google webmaster account that I set up, it has http://mysitename.com and www.mysitename.com as two separate sites. Each has it's own set of links and search rankings. I'm worried that I have some sort of SEO issue here.
I ended up setting www.mysitename.com as my preferred site name in google webmaster to deal with this.
Was that the right thing to do? Do I have an SEO issue here? Anyone have any advice?