I know that this is old, but none of the solutions offered worked for me and I wanted to share my experience.
I contacted Squarespace's support on two separate occasions but neither of the two agents I spoke with could help with it either, and both blamed Google for it.
The point is that the actual path (URI) to the resource is required by Google for the image to show up in the search results. Support told me that they do not disclose this kind of information, so I set out on a quest to find it myself.
This is how I did it:
(on Chrome) Open the dev console (F12);
Click on the three vertical dots on the bottom-left of the console to "Search..."
Search for the term "favicon"
You will find it someplace in the HTML, mine was line 10 but your mileage may vary: <link rel="shortcut icon" type="image/x-icon" href="https://images.squarespace-cdn.com/content/v1/62839a4f1318fc72022107b0/6dc4c692-1811-4d9a-9ced-1436a6d19390/favicon.ico?format=100w"/>
Once you find it, simply isolate the URI and then:
Go to the Squarespace editor
Click on the cog icon beside your homepage -> Advanced -> and enter the code as suggested by @TC_Dave before me: <link rel="icon" href="/path/to/favicon.ico">
My guess is that Google does not like the query string used by Squarespace and rather needs the simple URI, but I am a music producer and not a web developer, so who knows...