The phrase "square hole" means that something doesn't fit in or belong to a particular situation or group. It is like trying to fit a square shape into a round hole. Full definition
If you need us, we'll be at the bar, betting our friends that we can fit a large round coaster through a smaller square hole in a sheet of paper. (businessinsider.com)
This infinity scarf is made using four neon yarn colors, and has a pretty design with descending square holes. (12thblog.com)
Here's something you've probably never thought much about: Is there a way to get a round disk through a smaller square hole in a piece of paper? (businessinsider.com)