Aux outputs from mixers have most always have been line level mono. They can be balanced or not depending on the price point.
The intendend use is to send signals to monitor systems which are normally mono.
In a live situation those outputs would be driving a few floor monitors and the IE transmiters.
Some IE transmiters have mono and stereo input options.
A hardwired system will require a headphone amp as you say you have.
You don't mention what the brand or model the headphone ampe is so can't say what will be happening, but generally you get 8 mono mixes or 4 stereo. The chioce is yours. Mono is really all you need for recording.
If the headphone amp has only TRS headphone input(s), then you'll need so many "insert cables" 1/4" TS x2 to 1x TRS. If it only has one input then all you get is one stereo mix. But only guessing whithout more info about the amp.