have you checked the translation settings X ,Y & Z just to make sure they are set to 0
On the character that the eyes are popping out, the translation X, Y,& Z are not all set to zero. However, because of the character’s face setting each one to zero messes up the setting positioning of the eyes.
I am thinking that’s where the problem lies as the symmetry command is adjusting/ mirroring all translation settings its doing the same with the eyes.
Option 1 take note of the original settings and after using the symmetry command reset them back to what they where.
Option 2. Not sure if this will work find out if there is any way to deselect bone/nodes from the symmetry command before using it I have not played with it enough to know for sure.
that’s all I can think of but let’s see if any one else chimes in