Another way to fix your problem is to select each object and in the Editing Buttons (F-9) in the "Mesh" panel click the "Center New" button.
That's even better, and it's something I should've eventually discovered on my own. Actually, I kind of did, as I forged ahead and learned myself how to make the eyes track an object. Their initial origin is the same point as the rest of the figure, at or near 0,0,0... and "track to" induced all sorts of wildness 'til I caught on and applied "Center New".
Since I've spent much of my time here offering problems, I'll share a solution for a change! For those wanting the ability to pose the eyes, you'll want each eye to move independently of the other, which I imagine is difficult if they're "locked" together as part of the same object. What I did was to Duplicate the Eyes object, rename it "Eye.L" and move it to another layer. I renamed the current object "Eye.R", then went into edit mode, and deleted the left eye. I then switched to the other object and deleted the right eye, moving that object back to the main layer when finished.
I made sure to parent each eye to the Head bone, and then Weight-Painted the entirety of each eye to a factor of "1". They'll move with the head, but they aren't prevented from rotating, then "Track To" is applied.
Thanks again again, Penix1!