The post linked below has a video that clearly demonstrates how to remove recommended videos after a embedded Youtube video finishes in Squarespace.
http://www.simpleandsoulful.com/blog/2015/3/25/how-to-embed-a-video-onto-your-squarespace-site-the-classy-way
The key part I had been missing is that you have to insert the iframe code by clicking where the pointer is hovering in this screenshot from the post's explanatory video. (Sorry: grainy video --> grainy screenshot.)