Just what Logitech headset do y have? Do they say it will work with a Mac?
For rack0 tack0, can you provided documentation that
The latest mini supports the Apple iPhone headset and microphone, I.e. Both input and output on one jack plug.
That is not discussed here:
Mac mini (Late 2012): External ports and connectors
So it would appear that it is straightforward to use the iPhone earphones and microphone, thanks for the extra info.
What connectors are on your Logitech headset? If there are two 3.5mm jack plugs one for earphones and one for mic, if so perhaps this is what you require
Again what Mac Mini have you got, we are guessing what will work until we know that information.