Select circle and square and freeze their transformations to reset all transformations and rotations to 0. Now open attributes for circle and in limit transformations for translation limits enter the values as shown in the below figure. This limits translation of circle by axis. so we have limited circle to move from -2 to 2 in axis so that you cannot move circle out of square.

So when this is prepared lets make driven keys. Open driven keys editor which is in Animate/set driven keys/set. New window should pop up. Select eye ball and freeze it’s transformations. You don’t need to do that if all translations and rotations are 0 and eye looks in front. Now select eye ball and in driven keys window click on LOAD DRIVER button. Select eye socket model and in stack select Blend shape node and click LOAD DRIVEN. In driver section choose rotate Y and in driven section do as shown in the below picture. On left side blends shape node, on right all 4 blend shapes. When finished press KEY button.

Now first key is placed. Now let us set that to 0 rotations of eyeball and all morph targets will be zero. Now we have just doene for Y axis of eyeball. Now let us repeat the same steps for Z axis also. Put all eye rotations on 0. Select rotate Z in driver section of driven keys window. Base 1 and bas 2 for driven. And place key for neutral pose. Then rotate -20 by Z axis and move should be matching with blend shape of that position and place key. Same should be done for 20 degree by Z axis. If everything is setup correctly you should have fully driven eye leads by eye ball. Now lets make same thing for the eye to be driven by circle. Once again open driven keys window but now load circle as driver and eye as driven. For circle mark translate Z and for eye (driven) rotate X Y Z as shown in the below picture.

Place key for neutral position. Move your circle on one side and rotate eyeball on that side and place key. same for the other side. Then change in driver on Y axis place key for neutral position. Move circle up and rotate eye to look up and place key again. Then for down side do as explained, that is it we have finished with eye rig. Now it’s time to test. If you fallowed all steps correctly it should work fine.