As your teacher say, the opposite meaning of ''なんでも食べる'' is ''なにも食べない'', the opposite of ''なんでもある'' is ''なにもない''.
Now what is ''なんでもない''!?
The expression of its opposite meaning is なにかだ or なにかである, though that is rarely used.
ない in the sentence ''なんでもない'' is not not to exist but just a negative form of 〜だ or 〜である. You can also say ''なんでもありません'' with politeness.
A. この店にはなんでもある。
(This shop has everything.)
B. この店にはなにもない。
(This shop doesn't have anything.)
A and B are spoken about what the shop has. ある and ない in these sentences have the meanings of to exist or not.
When can we use ''なんでもない''?
If your friend looks disappointed and blue, you would ask him or her what happened and say ''どうしたの?''.
S/he can answer ''なんでもない'' if s/he doesn't mention what happened on him or her, or s/he doesn't want to make you worry.
(added)
I want you to think about ''だれもいない'' and ''だれでもない'' too.
''だれもいない'' is Nobody exists. or There is nobody. It's the same usage of ''なにもない''.
While, ''だれでもない'' is It's not anyone. This is the same usage as ''なんでもない''.
Please notice that not いない but ない is used in ''だれでもない'' even though ''いる/いない'' (not ある/ない) should be used for persons. That's because the ない in ''だれでもない'' is not not to exist but just not, is the same as ない in ''私{わたし}は学生{がくせい}ではない''.
So both ''なにもない'' and ''なんでもない'' are grammatical.
- なにかがある。Something exists.
- なにもない。Nothing exists.
- なにかだ。(the same as なにかである) It's something.
- なんでもない。It's not anything.
- だれかがいる。Someone exists.
- だれもいない。Nobody exists.
- だれかだ。It's someone.
- だれでもない。It's not anyone.