Hi there, the method that Squarespace uses for pointing a custom domain is not supported here on WordPress.com. We do have a workaround for brand new domains that can't be transferred, but it requires a Site Redirect upgrade which expires annually. For that reason it's not recommended.
However, it doesn't need to take 5 days to get your domain registration transferred -- once you've initiated the transfer (as I see you have done) you can manually accept the transfer and skip the automatic 5-day wait.
The complete steps to transfer a domain are as follows:
1. Remove private registration and unlock the domain
2. Log into your domain administration area as explained here:
3. Send yourself an authorization code
4. Initiate the transfer at your new registrar and enter the authorization code
5. Log into your domain administration area in your WordPress.com dashboard again
6. In the domain administration panel, click on the menu link for "Domains" and select "Transfers"
7. Under Status, click "Accept or Decline"
8. Follow the remaining prompts to accept the transfer of the domain to your new registrar
I note that your domain is already showing as pending transfer, so you can start from step 5 above.
Please let me know if you get stuck!