I'm pretty sure the outcome of this hinges upon whether or not that was part of his regular job duties.
For example, at one time (as a JW) I had an evening job in a small hotel on the beach. I woprked at the front desk but one of my responsibilities was to walk around the pool area and bldg just before sunset (since the mainenance man would have left by then) and make sure all the lights were coming on, for the pool, the pool deck, parking lot and so on. If not on I could simply check the circuit breakers or timers and flip them on. If there was a problem I could call the mantenance man to come back. During the holidays, Christmas lights were put up. So, during that season I would also have to flip a few extra switches as I made my rounds, to turn on the Christmas lights. I did not consider that as "participating" in the holidays, so I went ahead and included that in my routine. However, if I had been asked to help put up the lights or to decorate the tree in the lobby, that would have been different, because it was never part of my job to change any lightbulbs or hang any signs. My main job was at the front desk.
The WT used the example of a witness who works at a cashier in a store that sells many items. If a customer comes thru the line with many items, and among those items is a pkg of cigarettes, that witness could in good conscience ring up the cigarettes along with the other items. (we can add to this illustrations a JW who is a Walmart cashier and a customer is buying holiday items or birthday cards.) However it would not be appropriate for the witness to work in a store that sells only cigarettes and related items (a tobacco shop).
What if a JW sister had a job cleaning houses for worldly people - as many do - and one of her duties was to dust all the knick knacks on the shelves and tables. The one day the householder adds a religious statue or a Santa Claus figure. Would the witness dust everything in the house except that one item?
So if this man's job as a mechanic included tasks such as sweeping the tarmac, turning on outside lights, raising and lowering the flag each day, that should be addressed when applying for the job. If he accepted the job on the condition that this was one of his responsibilities, then he is in the wrong. However, if he had never been asked to handle the flag before, all the time he had worked there, and the boss suddenly wants him to do it - knowing that the employee had previously explained his beliefs - then the boss was wrong.
However, in the big picture, the employee should be "reasonable". What if the boss had asked him to put up a flag or sign that was for advertising, or that said, "Now Open" or something like that. He would have done it. So, putting up the national flag wouldn't requrie any more effort than that. He would be just hanging up something for other people to see. He himself would not be worshipping it or saluting it.
I think some witnesses just use this non-participation thing as a way to become a martyr unnecessarily, thinking they'll earn brownie points with the organization.