Are there words or phrases that I can substitute for the phrase "single-sex schools"? I need this because I do not want to keep repeating the same phrase throughout my ten-minute speech.
I applaud you for being concerned about wearing your audience out by being too repetitive, though of course synonyms for the phrase "single-sex schools" are limited. There is no word which specifically means "boys' school" or "girls' school" (though I think you might have been searching for Eton in the question I had to delete).
The only other phrase which means essentially the same thing is "single-gender schools." For some variety, however, you can exchange the word schools for education in either phrase: single-sex education and single-gender education. It is possible to use "same-sex education," but this phrase is potentially confusing and likely to trigger other responses in your audience. It would probably only serve as a distraction if you used it.
Another way to think about this dilemma is to find ways to avoid saying the phrase at all. For example, once you have established the concept of single-sex schools, you can refer to them as an alternative to traditional schools. (as in "this alternative to coeducational (traditional) education is a viable option because..."). As long as your audience clearly knows what you are comparing, you will find many opportunities to avoid repeating the same phrase as much as you might think.