That's not entirely correct either. Observe:
She was silent, and he was silent, too.
Yes, it is correct, but okay, yes, I didn't explain well.
She was silent, and he was silent too.
That's positive+ positive. You can tell that precisely
from the use of the word 'too'. There's one, let's say, 'state-of-affairs that is the case', and it is the case for both people. In this respect, 'too' and either are used in the same way.
In these constructions, you sometimes can't tell from the first clause whether it's meant as positive or negative. You need following context to know.
She was silent
Is that positive or negative? Did something happen or not happen? We don't know.
She was silent, and he didn't speak, either. [negative + negative]
She was silent, and he kept quiet too. [positive + positive]
Now we know.
"Either" is used to show that a negative statement("I didn't speak a word(=my being silent/not speaking)" is also("She sat silent(=her being silent/not speaking)") true about him.
Yes, the clause before 'either' must be a negative statement, in the sense that it needs a negative word, but not necessarily 'not' or 'never'.
He's starving himself, and she's barely eating, either.
The idea is that you're saying that for both people it's
not the case that they're eating properly. That's what 'negative' means with this.