No, God does not heal everyone.
Even for the Apostle Paul, who had the gift of healing, God did not heal him and remove the thorn in his flesh.
And lest I should be exalted above measure by the abundance of the revelations, a thorn in the flesh was given to me, a messenger of Satan to buffet me, lest I be exalted above measure. Concerning this thing I pleaded with the Lord three times that it might depart from me. And He said to me, “ My grace is sufficient for you, for My strength is made perfect in weakness.” Therefore most gladly I will rather boast in my infirmities, that the power of Christ may rest upon me. Therefore I take pleasure in infirmities, in reproaches, in needs, in persecutions, in distresses, for Christ's sake. For when I am weak, then I am strong.
2 Corinthians 12:7-10
The miraculous acts of healing performed by servants of God in the Bible are recorded by the Holy Spirit for one reason, to cause us to turn to the Great Physician, Jesus Christ. When we experience a healing from God, praise Him! But if we do not, praise Him! God uses illness for many reasons. In some cases, it is to demonstrate the Glory of God in one’s life. In others, it may be because of sins. In still others, it is to teach us humility and for us to be more dependent upon God. Joni Eareckson Tada became a quadriplegic after a tragic swimming accident at the age of 17 years. The next years of her life were filled with bitterness and disappointment. But slowly, God reached her heart and showed her how this accident was for His glory and her eventual joy. Many years later, Joni could proclaim, “God permits what He hates to accomplish that which He loves.”
The ultimate answer for why God does not heal everyone will only be known when we are in His presence. If you have accepted Jesus Christ as your Lord and Savior, God is preparing your life to serve Him, in this life and for eternity. If you have an illness, seek God’s will first and ask Him how He is using this illness to transform your life to make it more like Jesus.
Love and trust the Lord; seek His will in your life.