Dear teachers,
Swan said at the beginning of a sentence, only nobody and nothing are used.
It means;
Anyone can't do it. (X)
No one can do it. (O)

This rule is always true in English?
If so, is there any explainable reason that foreigners easily understand?

In addition, how about the following sentences?
Are they also incorrect and "any" should be replaced with "no (none)"?

1) Any of them didn't come to the party.
2) Any other girls are not as pretty as Jane.

This is an important question, Yun, and I'm glad you've brought it up.

When many people learn English, they get confused about when any or a word containing any- is negative or not negative. The simple truth is that any or a word containing any- is only negative when it comes after a negative verb. Here are examples:

I don't have any time to help you now.
She didn't know anybody at the party.

As you can see, because the two verbs in these sentences are negative, any and anybody carry a negative meaning, too.

But when we use any or any- at the beginning of a sentence (without a negative verb before either of them) or when we use them in a yes/no question, they don't have a negative idea. They mean "this one, that one, or another one, etc." In other words, any and any- refer to somebody or something that's not specified. Here are some examples:

Anybody who believes what he says is crazy!
Any time you need help, just call me, okay?
Do you have any questions?

Let's look at Swan's example again: Anyone can't do it.
The reason this is ungrammatical is that the sentence begins with anyone, so it isn't negative in meaning; it means "this person or that person or another person, etc." But the verb is negative, so there's a conflict here. That's why Swan changed anyone to no one. And since the sentence now begins with a negative word, the verb can't be negative too since we don't use double negatives in English, so can't has to be changed to can.

I hope this explanation is clear enough for you, Yun.

As for your sentences, we need to make the same kinds of changes that Swan made in his example:

1) Any of them didn't come to the party. --> None of them came to the party.

2) Any other girls are not as pretty as Jane. --> None of the girls is/are as pretty as Jane.

Now, let's go one step further. If we change the two statements above into yes/no questions, we can use any:

1) Did any of them come to the party?
2) Are any other girls as pretty as Jane?

Thank you for the detailed explanation!