My tenant states that they don't have to gjve 30 days notice to move out in New Jersey claiming it is not in writing
They do not have a written lease

Their legal notice has to be the same time frame as the way they pay rent - if rent is due on the 1t of, say, August you must get notice BEFORE the first of July.

That would be one month - and it's not figured in calendar days.

They may have no written lease but they are governed by state code. LSNJLAW - F: Chapter 5: Ending or Breaking Your Lease