Neither am I / me neither in a specific context

A friend of mine and I were talking about a party that we've had today and he told me the following:

I won't have dinner at home, lol.

And I replied:

Neither am I.

Based on these topics, Using either, neither and too and Is the statement "Neither am I" right?, my answer was not wrong at all in myself comprehension.

I want to know your opinion, is the above wrong? If so, why? And what's the correctiest way to express equality in that situation?

To:

I won't [=will not] have dinner at home, lol.

The correct answer should be:

Neither will I.

But:

I'm not [=am not] in a mood to go out for dinner tonight.

Neither am I. I think I'll just stay home.

• Another option that works for both examples is "me neither."

I won't have dinner at home, lol.

Me neither.

And

I'm not in a mood to go out for dinner tonight.

Me neither. I think I'll just stay home.