Your question is a very interesting one, that many of us have either not thought about or have taken for granted. Nevertheless, I would submit to you that this is a question that has arisen out of a couple of common theological misunderstandings. In fact, I would say that it is a marvel that so few have asked your question when so many hold to the same error in understanding.

I would agree with you that the Scripture speaks of God hating the wicked. However, I do not think that this implies that God hates the elect before their conversion. The elect are chosen and predestined in love, in the Beloved (Eph 1:4–5). Before their conversion, they are “children of wrath” (Eph 2:3), but certainly, they are not hated of God as many believe. When Paul says that “we were enemies” (Rom 5:10), I do not think it means that God took us as His enemies, but that we were at enmity towards God (cf. Rom 8:7). I hope you can see the bearing that this point has on your question. If it is true that God hates the elect before their conversion, then unconverted elect may be treated as reprobate, and there is some basis for us to question whether we may pray for them (cf. Jn 17:9; 1 Jn 5:16). But if God always loves the elect even before their conversion (treating them like the Prodigal son, cf. Lk 15:20), then there is no way we can conclude whether an unconverted person is hated or loved of God, and therefore 1 John 5:14 is simply not applicable, seeing we do not know who is elect and who is not.

But putting aside the theological argument, we see that the Scripture has a number of examples of praying for unconverted persons, as well as instructions to pray for them.

Consider, firstly, the example of how the prophet from Judah prayed for Jeroboam, a wicked king (1 Kgs 14:9–11), and how the Lord restored his shrivelled hand in answer to the prophet’s prayer (1 Kgs 13:6).

Secondly, note how the Lord Jesus teaches us:

Love your enemies, bless them that curse you, do good to them that hate you, and pray for them which despitefully use you, and persecute you (Mt 5:44).

While the Lord does not say that those who despitefully use us or persecute us are unbelievers, how many believers would do so to Christians?

And thirdly, the Apostle Paul instructs us:

I exhort therefore, that, first of all, supplications, prayers, intercessions, and giving of thanks, be made for all men; For kings, and for all that are in authority; that we may lead a quiet and peaceable life in all godliness and honesty. For this is good and acceptable in the sight of God our Saviour (1 Tim 2:1–3).

Now, there is no doubt that Paul has in mind particularly unbelieving magistrates, since he adds: “Who will have all men [i.e., all classes of men] to be saved, and to come unto the knowledge of the truth” (1 Tim 2:4). Is Paul teaching us only to pray for them in regards to work of government, and in regards to their salvation? Can we not pray for their recovery from illness? Death would not only interfere with the ability to rule, but could cut off all opportunity for their salvation.

Base on these points, I would conclude that it is not wrong for Christians to pray for the unconverted pertaining to their health. However, I think that as long as we are praying for an unconverted person in anyway, we should pray for his salvation, for this is the greatest need of an unconverted person. In fact, in as far as we do not know who the elect of God are, we ought to be very fervent in prayer for an unbeliever we know, who is suffering from some severe illness that may lead to death. Relief from pain, of course, should be one of our petitions, especially when we are praying for a loved one or praying in the hearing of the unbeliever. But along with such a petition, we should pray that the person be spared so that he will have many more opportunities to hear the Gospel and to repent and believe in the Lord. And not only so, but we should also petition the Lord to open the eyes of such a person that he may be brought by the experience of physical misery to think about eternal misery, that he may be shut up to Christ.