1) CloudFlare is flakey. If you're attempting to manage SEO, avoid using CloudFlare or any tech which lives between your Website + your visitors.
2) Squarespace allows very little SEO control. Switch to WordPress, if you're running a business you'll be working on for years.
I don't agree 100% with David's answer, but I'm not the server expert that he is. Instead, I'll answer from a purely Google-centric point of view.
Generally speaking, faster is better. If that means achieving the speed through CF, then that's fine. Google doesn't care where your website is hosted (other than bad neighborhood/webspam signals). Which is to say: SS vs. CF vs. any other standard host is irrelevant for SEO.... at face value.
Where there start to be big differences is in speed and security.
Given that it sounds like you've already chosen SS, I'd say sure, go ahead w/ CF... or any other proxy/CDN, **if** it's reliable and fast.
At the end of the day, Google wants to send its users to great websites with great experiences (because that makes the user feel good about using Google).
So if CF or another CDN helps you deliver a fast, reliable, secure experience for your users, then Google is happy.
One last disclaimer/reminder: this answer (and my expertise) do not take server/DNS/CDN technology into account. There are always upsides and down. SEO is only one seat at the decision-making table. YMMV, but if it were me, I'd vote yes.
I started my 20 year career in the hosting and ISP business. I believe you are asking if there are negative consequences to adding a CDN like cloudflare to your domain. I would say there are only benefits. These days, the performance of your website is a ranking factor. Having the site responsive to requests regardless of where in the world they come from, can only be a benefit. My experience of Squarespace is that it hasn’t kept up with the innovations in CDN and the great things that Cloudflare have been doing. If you want to chat further, let me know!
As a general rule, no. However, there have been cases where switching a site to CloudFlare has seen significant negative SEO impact, presumably because of what is often referred to as "bad neighbors" or IP tainting. Basically, someone with poor/unlisted content sharing an IP address with the site in question. There is no way to tell for sure. Then again, I am not convinced there will be any benefit for having CloudFlare in front of SquareSpace domain either. Might not be worth the hassle.
