I'm not really feeling the whole "2 pairs of eyes stacked together" look lol

If you go under the "Avatar" section of the wardrobe, you'll be able to change the mouths, eyes, and nose for each expression, including opting to have no eyes and no mouth. Once you make an expression with no eyes, you can choose that expression with the dropdown under Basic Traits! After that, just click save and use, and once the cache updates, you should be wearing that expression and thus have no default eyes!

You can quickly change expressions under "Home", just make sure to save any expressions you edit within the avatar section. 8)