Use first-generation student(s) on first reference. First-gen is acceptable on second reference. Use a hyphen when the term is used as an adjective. Ex: She is a first-generation student. Ex: She is first generation.
The University of Missouri defines a first-generation student as one whose parents do not have a bachelor’s degree from a college or university. This also includes students who, prior to age 18, did not live with or receive support from a parent who has received a bachelor’s degree.