None of these answers is correct.
Favicons are not always "favicon.ico" in the root. That was true in the 90s.
Squarespace support for custom favicons is broken. The template code (in the header element, it's a link with rel="shortcut icon") always uses favicon.ico as the name of the file. This is wrong. You need to change the filename when you change the icon so that the browser knows to invalidate its cached version. Annoyingly, I have yet to find a way to remove the reference to favicon.ico from the Squarespace templates. Adding a second entry using the Code Injection page and pointing to another file I've uploaded works for Chrome (which happens to use the last icon reference it encounters) but not for IE (which happens to use the first, apparently).
It's also annoying that Squarespace is full of incorrect statements like "IE won't display PNG favicons." This is clearly untrue, but what it likely means to say is "old versions of IE (and other browsers) won't use PNG favicons."