To disable YouTube's "Related Videos" from showing up when using a YouTube URL, add ?rel=0at the end of your URL.
Before pasting into Squarespace, add ?rel=0 to the URL:
Note: You can paste the original URL into the Squarespace Video Block FIRST, and then add your ?rel=0 at the end, but make sure that Squarespace "refreshes" the video in your post. If it doesn't refresh, Squarespace will not pick up your change (adding the ?rel=0). A quick trick I do is paste in your normal URL, then type ?rel=0 at the end of the URL, then "Select All", "Cut" and "Paste" into the box. This removes the URL from the Squarespace Video Block, which removes the video in your post, then adds in the correct video with the disabled Related Videos code.