I have a 1099-MISC from a company that is based in Illinois, although the branch I work at is in NJ, where I reside. (It is DeVry; their headquarters is in Illinois but I work at a NJ campus). The boxes at the bottom of the 1099 that are for NJ are empty. Do I need to file this on my NJ State Tax Return? If not, do I need to file in Illinois? The 1099-MISC is for $5,400.

It is NJ income so report to NJ. It is not IL income.
If you are present in a state, then any income earned during this period, must be reported to the state.

If you live and work is a state but your employer is from another state, then your income is only subject to the state where you live and work. Read: Your U.S. Tax Return: Working or Living in Two or More states