Is this a correct sentence?

There is nothing wrong with your sentences.

But the short rejoinder would be "I don't either."
He doesn't like pizza. I don't (like it) either.

Is it correct to say, I like either. Here either is used as a direct object right?

tenjingIs it correct to say, I like either.

No. "Either" is only used in the negative, when it is an adverb, meaning, also.

Would you like a green apple or a banana?

I'll take either one.

Can't we say, I will take either. Like,
A. Which juice would you like to have? I will take either.(just either)

Or I will take either of them.right?

Can't we use 'either' as a subject and as an object? Like,
A. Either is fine.(instead of 'either one')
B. Which one should I take? You can take either.(instead of 'either one')

tenjingCan't we use 'either' as a subject and as an object?

Yes. In my personal conversations, I would say "either one."