Shhh.... please calm down. There is no need to shout. We only usually use 'engaged' in specific contexts, such as when you're talking about public toilet cubicles, which may say 'engaged' or 'vacant' on the door locks. But you can also say 'free' or 'not free' in this, or any other, context. If in doubt, use 'free' or 'not free', because it's a more flexible term which is more likely to be right.

In American English "NOT FREE" sounds less awkward than "ENGAGED"; however, I would think that "UNAVAILABLE" would be more common. I don't know how British English would handle it.

Workers in this office were all engaged in their task assigned by the chairman of the board.