In one of the Japanese classes I attended, I've been taught that while we use 「何も出来ない」to say "He cannot do anything", to say "He can do anything" we use 「何でも出来る」 instead of 「何も出来る」.
Why is there a grammar rule that says 「何も」 is used before negative predicate while 「何でも」 is used before positive predicate? Why do we need additional で particle for the positive predicate?
Is the rule still being followed, and are there any exceptions (something like exceptions to the 全然+negative rule)?
Rather than memorizing edge cases like this one, I think the key here lies in understanding the difference between も and でも in this context.
In positive statements using も, the grouping is explicit. In other words, when you say 何も, だれも, どれも, and so on, it's clear through context or prior statements what "every" includes:
ピアノ、ギター、ドラム…彼はどれ｛○も／×でも｝上手に弾ける。 Piano, guitar, drums…he can play all of them well.
友だちはだれ｛○も／×でも｝DSを持っている。 All my friends have a DS.
私がお店に入ったとき、彼女はいつ｛○も／△でも｝いる。 When I enter the store, she's always there.
In negative statements using も, you don't have to worry about qualifying the scope of the statement, so 何もできない and だれもいない are sufficient.
With でも, however, the grouping can be left unstated:
こんな簡単な問題はだれ｛△も／○でも｝解けるよ。 Anyone can solve a simple problem like this.
これで本をいつ｛×も／○でも｝読める。 With this I can read a book at any time.
娘は好き嫌いが全然なくて、何｛×も／○でも｝食べてくれる。 My daughter isn't picky at all, and she'll eat anything.
Every language has lexical items which are restricted, seemingly arbitrarily, to specific contexts. Let's start with some English sentences to introduce the concept:
1a. I don't like her at all.
1b. *I like her at all.
The first sentence is fine, but the second sentence doesn't work because at all is a Negative Polarity Item (NPI)—a lexical item restricted to negative contexts.
Sometimes we have pairs of positive and negative polarity items:
2a. I like her, too.
2b. I don't like her, either.
Here, too and either are a positive-negative pair. If we swap them, the sentences don't work (at least, not with the same meaning):
3a. *I like her, either.
3b. *I don't like her, too.
Many NPIs can also appear in interrogative contexts—that is to say, in questions like 4c:
4a. I don't like her at all.
4b. *I like her at all.
4c. Do you like her at all?
But every polarity item has its own requirements, which can be arbitrary, complex, and context-dependent. For example, till is an NPI in clauses with a punctual interpretation:
5a. We won't leave till six o'clock.
5b. ??We will leave till six o'clock.
In 5a we interpret leave as a punctual event, one that takes place in a single point in time, but till can't appear in positive punctual predicates, so we can't interpret 5b the same way. Instead, we interpret 5b as durative—it sounds like we're planning on leaving for a while, possibly for hours at a time. This is, of course, nonsense, so 5b is highly questionable.
English is full of examples like these, and unsurprisingly, so is Japanese.
Japanese has a number of NPIs. One of the first ones taught to learners is しか, which is always followed by some form of negation:
6a. 行くしかない！
6b. *行くしかある！
There's no reason in principle why this is true. Positive しか could make sense, and indeed 日本国語大辞典 notes that there are rare historical examples of しか unaccompanied by negation. But in Japanese as it's spoken today, the negative requirement is more or less an ironclad rule.
Unlike English, Japanese is verb-final, and negation usually comes toward the end of a sentence. Many Japanese NPIs appear early, and they're often signals that a negative predicate is coming up:
7a. 決して忘れたりしない。
7b. *決して忘れたりする。
In many cases, this signal is strong enough that you can leave out the actual part of the predicate containing the negation (as long as it can be inferred from context):
8a. ちっとも変わらない。
8b. ちっとも。
This is true even for items like 全然 which have an association with negatives rather than a strict requirement:
9a. 全然わからない。
9b. 全然。
(Of course, I haven't supplied any context here; you'll have to imagine the context that allows the verb to be left out if you want 8b and 9b to make sense.)
Your example too is among the Japanese NPIs:
10a. 私は何も食べない。
10b. *私は何も食べる。
There's no reason in principle why 何も shouldn't be possible in positive contexts. It would make sense, just like certain other combinations of question words plus も. For example, どれも is fine either way:
11a. どれもおいしかった。
11b. どれもおいしくなかった。
So is いつも:
12a. いつも面白い。
12b. いつも面白くない。
So is 誰も:
13a. 誰もが知っている。
13b. 誰も知らない。
Notice here the arbitrary and quite strong preference for が in 13a. Notice too the arbitrary difference between 知っている and 知らない, using 〜ている only in the positive. Every word has its own story to tell, and its own way of being used.
And 何も has its own story, too. Unlike these other question words in examples 11-13, 何も happens to have an arbitrary restriction to the negative, while 何でも does not—and you'll just have to memorize this difference.
To convince you that it's truly arbitrary, consider あまり／あんまり. The following examples are borrowed from a paper by Ai Matsui:
14a. *今日はあんまりあつい。
14b. 今日はあんまりあつくない。
In this particular example, it seems that あんまり requires a negative context. However, this is not always the case. In the following example, it's licensed in a positive context by the presence of から:
14c. 部屋があんまりあつかったからエアコンを付けた。
Why? No reason. That's just how the language is.
Sometimes we can speculate as to why. Sometimes we can look at the historical evidence and try to find a reason from etymology. For example, some linguists suspect that しか contains a contracted version of the particle は, often associated with negative predicates. And it's true that we have relatively few cites for しか in positive contexts, even historically.
But generally speaking, why is a difficult and possibly unanswerable question when it comes to polarity items. For us learners, they're just something we'll have to memorize.
In this answer, the * symbol marks a sentence as ungrammatical.
This is not a very helpful answer, but: I think you are needlessly complicating a fairly simple grammar rule by looking for an "explanation". There might be some deep and obscure etymological link between the 'も' and 'でも' of 何も/誰も/いつも, but knowing it won't further your understanding of the rule itself.
It would be exactly the same as asking "why is there a grammar rule that says 'any[thing/one/time]' is used with a negative predicate while 'some[thing/one/time]' is used with a positive"...
As you can see, the analogy even holds up for the similarity between the words.
While 何でも出来ない sounds a little strange (but not unthinkable), negative forms with 何でも definitely exist. For example: 何でもないよ。 ("It's nothing").
Note: 全然+positive that you mention is not a "grammatical exception" (at least not yet). As the answers told you, it is a very colloquial use, strongly discouraged in anything more than casual conversation with friends.
