I have followed a few discussions about whether "on lunch", as opposed to "at lunch", is correct.
One opinion stated that "on lunch" indicates a status (on leave, on break) while "at lunch" is more likely conveying a sense of a person eating in a physical location.
Thus, I do not know whether "on lunch" is grammatically correct or it is just an informal, shortened version for "on lunch break".
Best Answer
For a work context:
To be on a break, to be on a lunch or tea break, but: to be at lunch.
In short, the guy's out of the office.