My opinion is different than those expressed above:
No, you should not write about “coming out.” You should write about two things: how you can help the school (two-thirds), and how the school can help you (one-third).
I’ve read and posted on the prep-school board for more than three years now. Plenty of times I have read postings from young people who stressed LBGTQ+ identity and ended up not getting in anywhere. And nobody I can recall who earned acceptances claimed to stress LBGTQ+ identity.
That’s very different from saying there are no LBGTQ+ students at boarding schools, including the GLADCHEMS. There are many. And I know that at my child’s school, it’s a nonissue among the student body. Yes, the school requires readings on LBGTQ+ identity here and there, but besides that it seems to be a nonissue either way.
Sexual identity for a young person can ebb and flow. Focus instead on family, school, relationships, community and striving to achieve, and you’ll likely be better off, and likely to have more compelling essays, as well.