何でも, 誰でも and so on are usually associated with non-negative predicates, whereas 何も, 誰も and so on are usually associated with negative predicates.
- 誰でもできる。 Everyone can do it.
- 誰もできない。 No one can do it.
- 何でもよい。 Everything is fine.
- 何もない。 There is nothing.
You can use safely ～でも with some apparently-negative-but-semantically-positive expressions such as:
- 私はいつでも構わない。 Whenever is fine for me.
- 誰でも問題ない。 There is no problem whoever it is.
- どこでも気にしない。 I don't care wherever it is.
In complex sentences involving more than one clause (i.e., there are more than one verb/predicate), ～でも and a negative expression can coexist in one sentence, of course.
- 誰でもできるわけではない。 It's not that everyone can do it.
- いつでもよいとは言わなかった。 I didn't say whenever is fine.
And you can say something like these.
- 何でもない。 It's nothing. / Don't worry. / Never mind. (lit. "It's not something")
- ほかの誰でもなくて、あなたが必要だ。 I need no one but you. (lit. "Not someone but you are necessary")
The last one may seem confusing. But the difference is similar to the difference between ペンはない ("There is no pen") and ペンではない ("This is not a pen"), so you can guess the meaning. See: Difference Between "何も" and "何でも"
