You said you knew the scriptures... so, calm down, and he's your answer:
James 5:14-15 (KJV)
Is any sick among you? let him call for the elders of the church; and let them pray over him, anointing him with oil in the name of the Lord: And the prayer of faith shall save the sick, and the Lord shall raise him up; and if he have committed sins, they shall be forgiven him.
This is actually the passage where we find recognition of the sacrament of the Anointing of the Sick. (Catechism paragraph 1510).
Prayer for the sick is extended out to everyone, whether they are of faith or not. However, this specific sacrament (not just praying for the sick, but the performance of this specific one), there are circumstances where it can be applied toward those not of faith, but it really does depend -- otherwise it really is only received for the faithful.
I'm sure there are disagreements on this one. At the very least, note that I didn't say prayer for wellness for the none-faithful isn't to happen -- we are always supposed to pray for this, provided it is God's will they get well.