Can a non-believer be prayed for especially - healing and why?
The Bible certainly advocates praying for non-believers for various reasons. “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.” (1 Timothy 2:1-2) Paul includes “all” men and rulers who may not be Christians as proper to pray for.
Although there is no other specific scripture saying what to pray for when praying for non-believers, I would certainly recommend praying for those that you want to become believers. I also see nothing wrong with praying for the health of non-believers. Particularly, it would be good to pray for their health so that they have further opportunity to believe.