Q: Can you fire an employee without due cause?

A: Yes. NY and NJ States are “employment-at-will,” states.
If there is no contract to restrict firing (like a collective bargaining agreement) an employer has the right to discharge an employee at any time for any reason.

This also protects the employee’s right to resign.

An employer may fire an employee for “no reason.”
An employer may also fire an employee for a reason that might seem arbitrary and unfair.
The employee is equally free to quit at any time without needing to explain or defend that decision.