It looks like Squarespace lets you use your own domain name so if your URLs will remain unchanged you will be fine. But if that is not the case, and some or all of your URLs will be changing, you will need to be able to do 301 redirects to redirect the old URLs to the new URLs. If they allow you to do this the SEO impact will be temporary and minimal (although you should contact any sites linking to you and ask them top update their links).
If you cannot do 301 redirect then you will have SEO issues as any links to pages that no longer exist will essentially become useless. You will also have to wait for the search engines to re-index your site, find the new pages, and then add them to their indices. This will almost certainly result is less traffic (assuming you are getting traffic from the search engines). It's almost like starting from scratch, SEO wise.
If your site currently gets good search engine rankings and good traffic as a result of it, this move may be more harmful then beneficial if your site structure is going to change a lot. If your site gets little to no traffic from search and/or has poor rankings this move can help as you will be able to better focus on your content which is what you need to get good rankings.