何も vs. 何でも
The way I remember this (probably only works for me), was when I was watching Chobits, there was a book in the anime called 誰もいない町.
So drawing from that context:
- 何（なに）も + negative
- 何（なん）でも + positive
For example:
何も食べません。
I’m not eating anything.
何でも食べます。
I’ll eat anything.
Other examples:
どれも選びません。
Not choosing anything.
どこも行きません。
Not going anywhere.
誰でも参加できます。
Anyone can join.
どれでも選べます。
Can choose anything.
どこでも行けます。
Can go anywhere.
Exception: どこも
When describing a state, どこも + positive is used.
どこも混んでいます。
It’s crowded everywhere.
Reference
Reading this reference helps me make sense of the exception too. The grouping is like Everywhere—this place, that place, and that place—is crowded.