If I were to take just one semester off, would I need to reapply to NEIU to register for the next semester?
You can take up to two semesters off without having to reapply and be readmitted to Northeastern. For example, if you take off the Fall 2016 semester, you will need to register for the Summer 2017 semester otherwise you will no longer be an active student at Northeastern. So, based on what you have shared, it seems fine for you to take one semester off. You can also check with your advisor or Enrollment Management Services for additional information.