There is no name for the mascot. But it is a wildcat! The setting of the high school is in St. George Utah, The wildcat mascot is the symbol of the high school musical school, but is not mentioned much in the show.
The High school which toots the Unicorn as its mascot is New Braunfels High School located in New Braunfels Texas. The North Carolina School of Science and Mathematics in Durham, North Carolina also has the unicorn as its school mascot.