Since we don't know the gender of the student it should be "un estudiante". We always use "un" when the gender is unknown here. But if we know that it's a female student we use "una estudiante". I had an exercise where I was supposed to translate "a student" and the only correct option was "una estudiante". It is neither right or wrong in general, but wrong in this case since the gender in the exercise is unknown.
No, the spelling of estudiante doesn't change between genders:
el estudiante, la estudiante.
The answer was una estudiante among the choices, even though i picked un estudiante.