I have received several questions about the use of "whoever" in the translation of John 3:16, so I thought it would be good to clarify at least one thing.
Correct, the indefinite relative pronoun ὅστις does not occur in John 3:16, but language is not so monolithic that there is only one way to say something. In fact, whenever a commentary argues that if the author had meant to say one thing, he would have said it "this way," you should be suspicious. That's a naive approach to language.
However, we do have an indefinite construction in John 3:16 with the use of πᾶς and an articular imperfective participle (πᾶς ἡ πιστευών) used to indicate a generic, "general utterance" (see Wallace, 615f.). Just do a search for that construction and you can see it is universal in intent.
For example, "But I say to you that whoever looks at (πᾶς ὁ βλέπων) a woman to lust after her has already committed adultery with her in his heart" (Matt 5:28). Isn't Jesus saying this is a generic statement, true of all who look with the intent of lusting? Of course it is.
Interestingly, v 28 is followed by v 32 that uses another explicitly indefinite contraction. "But I say to you that anyone who divorces (ὃς ἂν ἀπολύσῃ) his wife, except on the ground of sexual immorality, makes her commit adultery" (Matt 5:32). ὃς ἄν is explicitly indefinite and general.
The first meaning of πᾶς in BDAG is "pert[aining] to totality with focus on its individual components, each, every; any." The second is "any and every." Sounds indefinite to me.
Contextually, John is asserting a relatively unusual notion that God not only loves those who follow him (John's normal usage) but he actually loves the entire world, hence requiring an indefinite construction. To limit the meaning of the statement to a subgroup of people, "those among you who believe," is to read in a theology not supported by the Greek (and I am Reformed).
In the larger context, it agrees with statements like 1 Tim 2:4 that says God "wishes all people (πάντας ἀνθρώπους) to be saved and to come into a knowledge of the truth."
True, each/every person who believes is a subset of the whole (the "world"), and the gift of eternal life is only for that subset, but to somehow limit God's love to a subset of people runs counter to the Greek, the meaning of πᾶς, the grammar, the immediate context, and the larger context. If you believe in election (as I do), then you understand πᾶς ὁ πιστεύων as referring to the elect, but let's not dismiss the clear meaning of the text and suggest that God does not, in some way, love the world.
Can you translate the verse without "whoever"? Sure, as long as you choose words that are not limiting. "God loved the world so he gave his only Son, that every one who believes in him will not perish but have eternal life."