Salam and thank you for your question.
The prayer using the eyes without moving any other body parts is meant for people who are sick and are not able to move around. To perform this prayer, a person must fulfill all other obligations required for salat (Islamic prayer) such as performing wudu (ablution) and not moving unnecessarily or talking to someone during the prayer as you already know.
However, in your situation you may do one of the following in order of preference:
1. Combine zuhr and asr by taking a break and praying in a place where you will not be jeopardizing your position at work.
2. If the above solution is not possible, combine zuhr and asr as soon as you leave your work, assuming that it is still before maghrib. You may do so in any place outside your work that you might find convenient.
3. If the above solutions are not feasible in your situation, then pray in any way possible by using your eyes only, as you have mentioned, but you must replace those prayers later at night when at home since a person without any disability may not pray the obligatory prayers without performing the ruku’ and sujood from a standing position.
Performing the prayer using your eyes will relief you from the time constraint of the prayer; repeating the prayer will relief you from the rest of the obligations. Ask your officers if you are allowed to meditate for few minutes and combine zuhr and asr during that time.
In the long run, you might want to find out how to wisely approach your government so that they allow Muslims to pray during a short break when on duty.