The definition of adverb from Webster's College (1) and Collins (2; quoted from definition 1a), via thefreedictionary.com:
- a member of a class of words functioning as modifiers of verbs, adjectives, other adverbs, or clauses, typically expressing some relation of place, time, manner, degree, means, cause, result, exception, etc. ...
- a word or group of words that serves to modify a whole sentence, a verb, another adverb, or an adjective
Modifying sentences as above is the function of even, thus it's an adverb.
It's not a conjunction because even doesn't group or join things together. It neither defines collections (e.g. A, B or C) nor connects two parts of a sentence (e.g. I went to the store and bought some things). I would even go farther and say it separates things, by using emphasis to make them stand out. A verb, quality or condition marked with even is notably different from the norm; even provides emphasis, not connectivity. You can have conjunction phrases which include even, such as even so, but in these cases even serves to modify the meaning of the actual conjunction. If you remove even from such a conjunction, it will still be grammatical (though it may no longer be semantically correct). Even doesn't join a clause to the rest of the sentence; it modifies the meaning of the clause itself.
Even may sometimes appear to function in a prepositional manner, just as it does in your example. But it's not a preposition, because it doesn't require a noun target (or any target, even) and it serves to modify a clause, not introduce one. In these cases, even stands at the beginning of an adverbial phrase. It's grammatical and sensible to use a naked, targetless even at the end of a sentence, but it's generally not to do so with a preposition. Observe:
- We all went, even Steve.
- We all went, even.
The two meanings are marginally different, but the general sense remains the same with or without Steve. Everyone was able to go, and this was unexpected or difficult to accomplish. Calling out Steve serves to indicate of the precise level of difficulty; presumably he is particularly unavailable or reluctant relative to the rest of the group.
No one can do it, even him.
And for my example above, what exactly do you call the boldface text ... ?
I'd call it a clause functioning as an adverb. The clause emphasizes the whole sentence (no one can do it), by using whoever he is as a marker of difficulty, just as with Steve in my example.
As an aside, this sentence is unsemantic, because he is a member of the group who are unable to perform the action. But the negative is applied to the subject rather than the verb, so this suggests he can do it, because no one is contrasted with him. Adding not before even will fix this problem, by keeping him inside the negated group.