Hi DanPro!
I’m afraid it doesn’t work. I’ve already done with this method earlier (parenting the eye mesh directly to the eye bone), despite the eyes still leap out of the head. When I grab and move the whole body or torso the eyes jump out and go to same direction where I carry to but their distance are exponential length for the body’s changed position.
Just curious… So you have a lattice on your eyes? I had one on mine, and when I removed the lattice (and the modifier), that problem went away. (There’s probably a way to do it with the lattice, but I haven’t figured it out yet.)
Also, double check the weight painting to make sure that they are fully parented to the bone you want them connected to.
EUREKA! I’ve found the solution of this problem. Not the lattice or something, but you have led me to adequate area with the tip around of weight painting. The problem was that the eye contained all attributes of the body in the vertex groups so I’ve removed all of them and everything is fine. Thank you very much! 
That means that you did not remove the armature modifier from your eye mesh. It’s no longer needed, and as you have found out, the vertex groups are not needed either.
Good luck!
Hi DanPro! You’re right! I overlooked it and I’ll remember it. By all means, thank you! 
