Neither your friend's question nor your version of it is correct; in both, the interrogatory pronoun should be the objective whom. Other than that, the two are both correct.
The construction in both sentences is an infinitive clause, consisting of a subject, an infinitive, and any modifiers. One possible source of confusion in both sentences is the fact that the infinitive clause is noncontiguous, with words not in the clause between parts of the clause.
In an infinitive clause, since there is no predicate (it is instead an infinitive), the subject takes case based on the clause's function in the sentence, not the subject's function in the clause. Because of this, the subjects of both infinitive clauses should be accusative, for both clauses are direct objects (You want whom to win).
Your revision made the infinitive that of a linking verb, which takes a predicate nominative (misleading name) of the same case as the subject. Because the subject (winner) is objective, the predicate nominative must be so as well; hence, whom.
With this replacement, both sentences would be correct, and they would have the same meaning.