Let's get this out of the way right now: mongooses' eyes don't actually glow a fiery red when they get angry. Kipling invented that whole thing specifically for "Rikki-Tikki-Tavi."
Okay, now that we've got the zoology out of the way, what does this odd eye pigment have to do with the story? As they say, eyes are the window to the soul.
We know that Rikki-tikki's eyes "[grow] red and hot [when] he is angry" (32). This fiery temperament might not be such a great trait in a different kind of story, if it led the protagonist to action without thinking. But in "Rikki-Tikki-Tavi," temper is a good thing: it leads him to action and makes him brave.
And then there are Nag's "wicked snake's eyes that never change their expression, whatever the snake may be thinking of" (23). This expert, unreadable poker face hints at Nag's foreign nature. No one can read a snake's eyes because the snake is distinct from anything the narrator would consider normal. Pretty creepy.