I remember that guy. He claimed the ambiant light created enough shadow on the mini already.
Anyway, I say "Hell yeah" but disagree with the rest of that choice.
You shouldn't paint difficult stuff just to show off that you can paint difficult stuff.
Painting the eyes makes the mini complete.
I haven't spent too much time with tabletop gamers, but whenever I have, people come by and lean in real close to look at the minis.
Like Callumrice said, if you get real close, you will see if the eyes are painted. If they aren't, the mini just won't look complete.
And the eyes don't have to be detailed. If they are small, sometimes all I do is paint the socket black or brown and put a tiny (pink or gray or bone colored) dot on the left and on the right. That is the simplest method I know, and on the smaller stuff (and I do this on 10mm minis) it makes a big difference.