Purpose: Metrics derived from eye-gaze-tracking and pupillometry show promise for cognitive load assessment, potentially enhancing training and patient safety through user-specific feedback in tele-robotic surgery. However, current eye-tracking solutions' effectiveness in tele-robotic surgery is uncertain compared to everyday situations due to close-range interactions causing extreme pupil angles and occlusions. To assess the effectiveness of modern eye-gaze-tracking solutions in tele-robotic surgery, we compare the Tobii Pro 3 Glasses and Pupil Labs Core, evaluating their pupil diameter and gaze stability when integrated with the da Vinci Research Kit (dVRK). Methods: The study protocol includes a nine-point gaze calibration followed by pick-and-place task using the dVRK and is repeated three times. After a final calibration, users view a 3x3 grid of AprilTags, focusing on each marker for 10 seconds, to evaluate gaze stability across dVRK-screen positions with the L2-norm. Different gaze calibrations assess calibration's temporal deterioration due to head movements. Pupil diameter stability is evaluated using the FFT from the pupil diameter during the pick-and-place tasks. Users perform this routine with both head-worn eye-tracking systems. Results: Data collected from ten users indicate comparable pupil diameter stability. FFTs of pupil diameters show similar amplitudes in high-frequency components. Tobii Glasses show more temporal gaze stability compared to Pupil Labs, though both eye trackers yield a similar 4cm error in gaze estimation without an outdated calibration. Conclusion: Both eye trackers demonstrate similar stability of the pupil diameter and gaze, when the calibration is not outdated, indicating comparable eye-tracking and pupillometry performance in tele-robotic surgery settings.