Neither & either

The first one is. The second one is clumsy and best avoided. Neither nor is a negative, either or isn't.

When I join a band I will either sing or play the piano.

You could say "my wife has forbidden me to either phone or write to my mistress".

Is it not possible to use "either" in negative sentences?

You can either (!) indicate the negative meaning by using a positive and then negating it with 'neither... [nor]':

My throat is so sore that I can (positive) neither eat nor drink.
Which of these apples do you want? I want (positive) neither [one].

Or you can use a negative and follow it with 'either... [or]':

My throat is so sore that I cannot (negative) either eat or drink.
Which of these apples do you want? I don't want (negative) either [one].

What about "I don't like either John or Alan"? Is it correct to say so?

Yes. Both of these mean the same: I don't like either John or Alan; I like neither John nor Alan.

Thank you for your kind help.