Hi Berriepie,
I actually think that your mesh eyes need to be able to use BOM (bakes on mesh). And then you need to make sure you are not wearing any Alpha layer for your eyes. BOM doens't go toghether with the alpha layer. If your eyes are able to use BOM, you can just wear it and the BOM will automatically hide your classic avatar eyes. But I found out for me that not all mesh eyes are working with BOM. They are just not enabled to work with BOM. So make sure you figure that out first, good luck <3