In theory the answer is yes but I haven't gone through Square to purchase a domain name, I use the web hosting company I want to use for this purpose.
That being said, When you purchase a domain name you own it but the registrar may be Square, or whomever they are associated with, and in the future when you want to move to a different host they will have to release the domain and 'approve' the transfer. Shouldn't be a big deal but it may take a few days to get all of the of the moving parts to come together.
A lot of times you can get really good deals on domain names from web hosting companies. You just have to shop around. I use BlueHost and they give you a free domain name for 1 year if you sign up for their hosting service. I think GoDaddy has a similar deal.
