is it Biblical to claim that 'real' or 'strong' or 'competent' Christians will never get sick or be subject to any disease?
No, that is not what the Bible says. At the same time we should keep a few things in mind:
1. While Christ was on earth, he healed thousands. Some came to Him for healing, but some did not. However, even those who did not were healed or resurrected. Daily miracles were a part of Christ's ministry on earth, so that they would generate faith in Him.
2. God gave the gifts of healing and miracles for apostolic times. The apostles did heal people (though every incident may not have been recorded). But the Apostolic Fathers (who came immediately after) evidently did not have these gifts, so they may only have been to help spread the Gospel at the beginning.
3.
James did not speak about healing miracles. Rather he provided Christians with a way to be healed as recorded in James chapter 5. However, too many churches have ignored or neglected to follow his instructions to the letter. He said that the prayer of faith SHALL heal the sick, but certain conditions needed to be met So the prayer of faith appears to be a certain way to be healed.
4. Christians are not guaranteed perfect health throughout their lives.
However, every believer can go directly to God and Christ for healing. Sometimes they are healed and sometimes they are not.
5. There are many healing miracles done directly by God daily in answer to prayers. Which means that the miracles are not through a human agency, but from God Himself.
"Faith healers" are generally "fake healers".
6. God has provided
an astounding array of natural healing remedies through herbs and plants. Quinine is a good example. It is extracted from the Cinchona bark. It is not only an an excellent antiviral in itself , but also in HCQ and other derivatives. It also has other medicinal properties.
7. At the same time He has provided human beings with
an amazing immune system to protect them from bacterial and viral infections. But people need to understand that proper nutrition, exercise, and rest are absolutely necessary to maintain good health. And taking unnecessary risks in sports and other outdoor activities is just asking for trouble. So is smoking, drinking, and drugs.
8. Paul says that partaking of the Lord's supper unworthily will lead to weakness, sickness, and even premature death.
For he that eateth and drinketh unworthily, eateth and drinketh damnation [condemnation] to himself, not discerning the Lord's body. For this cause many are weak and sickly among you, and many sleep [die]. (1 Cor 11:29,30)
9. John speaks of a Christian brother (or sister) who may also die prematurely. He calls it " the sin unto death" because it is related to sins not repented of. So we need to also keep this mind.
10. In general God bestows good health on the majority of people (saved and unsaved). But there is no guarantee that we will always be healthy, and never be subject to infections, diseases, or disabilities. At the same God has also provided genuine medical science and skilled doctors and surgeons to help people overcome these things.
But from God's perspective, the health of the soul and spirit is even more important. God will have all men to be saved, and come unto the knowledge of the truth.