I haven't been enrolling classes for like over 3 quarters now, does that make me no longer a student?

The answer depends on your particular school's rules.  If you still have all the rights of a student, especially the right to enroll in another course, then you are still a student.  If your school has removed your rights, then you are not.

If you are not enrolled in classes, then you are not a student. Did you tell the university you needed to take off for a couple of quarters?

You need to check with your school.  I would assume you will need to reapply to continue there. 

It depends upon the college.  At best, you are "inactive", but could start up again without any fuss.  At worst, you have been dropped from the school, and would have to reapply.

The easiest way to find out is to go to the college website.  See if your student account is still active.  If it is, check to see if it gives you a status.

UPDATE

I just saw your other note where you talk about your academic dismissal.  That means you are no longer a student there.  If you generally want to tell people that you are a student, go ahead, but don't mention that school.