The SC guidebook states that when the words "Either" or "Neither" is used alone (without or/nor), they only take singular verbs.

I'm not sure if the following sentences are right (especially the second):

Either of them is fine.
Neither of them is fine.

Any advice?

Yep - they sound funny but they're right! When you use either/neither alone, you are essentially saying either ONE / neither ONE (the "one" is implied).

So either ONE of them IS fine.
Neither ONE of them IS fine.