Rafael Figueroa, dean of college guidance at Albuquerque Academy, was in the middle of tutoring Native American and Native Hawaiian students on how to write college application essays when the U.S. Supreme Court ruled that the race-conscious college admissions processes at Harvard and the University of North Carolina are unconstitutional.

Earlier in the week, he told the students that they shouldn’t feel like they need to talk about their ethnicity in their essays. But after the June 29 Supreme Court ruling, he backtracked. “If I told you that you didn’t have to write about...