I recently departed from my company with the handshake agreement that the company would not challenge my claim for unemployment in exchange for the two plus months notice I gave. If I file that I quit due to stress and the employer confirms this, what is the chance of denial by the State of NJ? Thanks.

It is quite unlikely that a claim of quitting due to stress is going to qualify for UI whether the employer agrees to it or not. In the very, very large majority of cases, if you quit you don't get benefits.

It is POSSIBLE, but by no means guaranteed, that if you explain to the state specifically what it was about your job that caused the stress AND those causes are generally considered "good cause" to quit and receive benefits, then you might qualify. Considering that those causes tend to mean illegal behavior on the part of the employer, however, it isn't likely that an employer will agree to that.