I've recently asked the question to the ESRI help team and apparently it's an unreported bug. I've also had a few suggestions from some people on GeoNet but none have worked. So here it goes:
The problem I am having is that my "XY Event Source" labels do not work when I convert them to shapefiles.
When I click "Label Features", my points in the shapefiles display a bunch of question marks (?).
I am using the Pigiarniq font which I have downloaded on my computer.
Even when I change the "Text Symbol" from the "Layer Properties" to Pigiarniq font instead of standard Arial, it comes up as question marks (?).