If they function as the subject, a singular verb; if they're followed by a prepositional phrase then the preposition's object tends to agree in number with the verb, though, practice varies.Originally Posted by jirikoo
In the sentences directly below, either functions as an adjective, so no need to worry about verb agreement.
My comments follow the examples.
 Do you like Susan or Petra? I like either girl.
 Do you like Susan or Petra? I like either girls.
 Do you like Susan or Petra? I like either of them (btw: does this sentence really mean both girls?)
 Do you like Susan or Petra? I like either one.
either has two meanings:
(i) one or the other (That meaning you don't want);
(ii) each of the two (planes; e.g., sides, ends, etc.;
e.g.,There are trees on either side of the street. (That's the meaning you're going for, right?)
OK. Well, let's take a closer look at the semantics. Here's a short between a ticket agent and a customer. The customer wants to tell the ticket agent that she doesn't have a preference over one seat or the other:
Ticket agent: Do you want a window or an aisle seat?
Customer: Either seat will do. I don't have a preference.
Notice that either is connected semantically to preference. Now, consider these verbs here: want, like, need. They require a definite preference. Which is why  sounds OK, but it also seems awkward because either goes against the semantic grain of the sentence by expressing an indefinite choice.
 I like either girl.
like = a definite preference
either = an indefinite choice
Do you see the semantic incompatibility at work there?
Example  "I like either girls" is also semantically awkward. 'girls' is plural; 'either', singular. Try, "I like either one of the girls."
Example  "I like either of them" is semantically sound given meaning (ii), each of the two).
Also note that, 'them' is plural, yes, but it's the object of the preposition 'of'.
Now,  doesn't mean that you like both girls. It means, you don't have a preference over one or the other; therefore you like both.
Lastly, sentence  "I like either one" is the same structure semantically as "I like either girl". Just a little replacement there. The pronoun 'one', though, seems to sound better in that context than the noun 'girl'. The reason being, it's short for 'one of': "I like either one of them; I like either one of the girls." Here the pronoun 'one' defines a preference, but either remains indefinite.
Semantic incompatibility. That's why your example sentences kind of sound OK but seem rather awkward.
I'll leave the rest to you.
All the best.