Open your image in Photoshop. The photo we'll work with in this walkthrough is below.
The first step is to duplicate the image, so you can work with a copy and not the original. Open the Image menu and select Duplicate.
Close the original image and work with the copy going forward.
Go to Window >> Arrange >> New Window for [Filename] to open the file in another window. You'll then have the same file open in two windows.
Choose one of the files, and then zoom in on the picture as much as you possibly can, and then make sure the eye is in the frame as shown below.
Select the Eye Dropper tool from the left menu.
Use the Eye Dropper tool to select a color from the zoomed in picture. Ideally, the color should be a mix of gray and the eye color (as shown below).
Then select the Brush tool.
Create a new layer before you paint anything with the Brush tool!
Select the new layer in the Layer menu and then paint over the red parts of the eye with the Brush tool. Be sure to do both eyes.
Add a Gaussian Blur to this new layer. Go to Filter >> Blur >> Gaussian Blur.
When prompted, give the Gaussian Blur a 1.0 pixel blur radius as shown below.
Open the blending options for this new layer: Layer >> Layer Style >> Blending Options.
Go to the Blend Mode dropdown and select Saturation. Click OK.
Duplicate the layer you just added the Saturation to (probably called Layer 1 unless you renamed it). Be sure the layer is selected, then go to Layer >> Duplicate Layer.
Add a Hue Blend Mode to the new layer. Adjust the Opacity of the Hue as needed. That should leave the eyes looking pretty good.
Below is a view of the final product with one eye fixed and the other untouched for the purposes of comparison.