If the eyes are essentially overexposed (without recoverable highlights), then the very BEST you'd likely be able to do with a completely automated solution is to select them with the Magic Wand tool, then make them pure black.
BUT... That will look the image look a little odd/dead, and in examples like the one above, you'll have secondary troubles where the flare will extend into the surroundings and look pretty weird. To wit:
A bit of cloning and quick painting in of small hard-edged white reflections, however, can help a lot:
This was literally no more than 1 minute of work. You will likely have to bite the bullet and do some hand editing.
By the way, I selected the bright parts with the Magic Wand tool set to tolerance 30 and Contiguous, then feathered the selection 1 pixel, then dragged the white point of a Curves operation to pure black twice, to make sure there weren't remnant edges of the bright parts. That sequence could easily be repeated via an action, so your process could go something like this:
1. Select eye reflections with the Magic Wand tool.
2. Run your action to blacken them.
3. Paint little reflections with a small round, hard-edged brush.
4. Clone out any weird highlight extensions into surroundings.
That will take a minute or two per photo.
In the long term, investigate getting an external flash, to where the light is off-axis from the lens, and maybe bouncing it.