I think I get why she was embarrassed. She has a small tongue (which she has a complex about) and couldn't properly say the word "uni-sex" because of it. She was embarrassed to try and say the word in front of others.
Since this series is ending soon, you guys might like this series. It's basically the same premise. It's called Tonari no Seki-kun. https://mangadex.org/title/5009/tonari-no-seki-kun
So from what it sounds like to me is that he did something bad to her when they were dating, causing them to break up. We have no idea if the ex wants to get back together or not. Hell what he did might just be weighing on his conscience and wants to apologize.