Sqsuarespace injects custom code blocks using innerHTML in JS - and also serves pages as a SPA (SinglePageApplication). This is not a good combination - it prevents browsers from loading e.g. <script> tags. They should have used eval() instead - and it would most likely have been fixed once and for all.
The fix is relatively easy though.
Head over to https://your-site-here.squarespace.com/config/design/styles and turn OFF Ajax Loading in order to get the filemail scripts to work