1. 나는 먹지 않겠어요.
First of all, I'd recommend you to use '저는' instead of '나는' because the verb is a polite form.
* verb stem + 지 않겠어요 : It is from the form of negation " verb stem + -지 않다'. '겠' is used to show your future intention.
2. 나는 안 먹겠어요.
It is from another negation form of '안 + verb'. '겠' is added to the verb stem to show future intention.
3. 나는 안 먹을게요.
*verb stem + (으)ㄹ 게요 : It focuses more on actions or decisions AS A REACTION TO
or AS A RESULT OF what the other person says or thinks.
So, "나는 안 먹을게요. " means "I will not eat it if you don't mind (or if you want me to)."
4. 나는 안 먹을 거예요.
*verb stem + (을)ㄹ 거예요 : It is used for your definite plan.
So, "나는 안 먹을 거예요." means " I will not eat it ( no matter what you say or think)."
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