A proof, or a sign of someone who has supernatural and immaterial powers is through conducting miracles. A miracle is defined as 'a surprising and welcome event that is not explicable by natural or scientific laws and is therefore considered to be the work of a divine agency; a highly improbable or extraordinary event, development, or accomplishment that brings very welcome consequences.' What evidence is there that Jesus of Nazareth performed such inexplicable, highly improbable, extraordinary supernatural acts? Besides the eye witness accounts of the synoptic gospels, we can look at other outside biblical sources, some even hostile to Jesus, yet, unintentionally validate him.
The Synoptic Gospels:
The Gospels record miracles performed that fall in two categories; healing and controlling nature. All his miracles were performed freely and did not profit off his actions (unlike faith 'healers' today). Jesus' healings are undeniable. People who have been disfigured, born with obvious birth defects, and have clear diseases such as lepers where completely healed to the point that the healing itself was clear, obvious, and without visual question (unlike supposed claims of healing today). The questioning was more along the lines of disbelief that it actually happened in real life stuned by the sudden and drastic physical change. The gospel records details that only eye witnesses would have known about. Is there any record of witnesses outside of the bible? Yes.
Extra Biblical Accounts
Papias of Hierapolis is recorded by Eusebius as speaking with Phillips daughters, John the Evangelist, and elders of the church who were disciples of the Apostles such as Polycarp. He is recorded as knowing first hand people who have been healed and raised from the dead.
The Egerton Gospel fragment is another amazing find. It is known as Papyrus Egerton 2 and the author is unknown but the date of the fragment is first century. It details a miracle that is not recorded in the bible in the second fragment. In John's gospel, he admits this could be true (John 21:25). It also describes a miracle of curing someone of leprosy in the first fragment. Thus an unknown author recording specific details of miracles outside of what was recorded in the bible.
To ask if there were non-believers who witnessed these miracles is almost an oxymoron. A non-believer would not see them as miracles initially. If the event did not compel them to believe, then they would not see the event as a miracle at all. They would attempt to explain them as something else; which Josephus and The Talmud did.
Unintentional Support
The Talmud records Jesus as practicing sorcery. Why would they think of 'sorcery' if he was just teacher? He clearly conducted some inexplicable, highly improbable, extraordinary supernatural acts that they refused to accept as a miracle. Matthew 12:22-23 even records their attempted justification. Later mentions of Jesus in the Jewish writings (Babylonian Sanhedrin107b, tHul2:22-23, Sanhedrin 43a; cf. t. Shabbat 11.15; b. Shabbat 104b) continue this as their only way to explain how he did what he did. They inadvertently support his inexplicable, highly improbable, extraordinary supernatural acts in history.
Even Josephus can't bring himself to call Jesus a miracle worker but came as close as he can in an attempt at being honest and accurate in his writings of history when he states: "for he was one who wrought surprising feats". What was so surprising about his feats if he was just a teacher? He clearly did more than just stand on hill sides and give sermons. He conducted "surprising feats" and Josephus lived around a time when people knew others who witnessed these 'feats'.
Celsus, a greek philosopher who wrote in the second century and is quoted by Origen who wrote around 220-230AD. The work Origen is quoting was written around 170AD. Celsus was very anti-Christian and helped fuel some of the anti-Christian sediment and persecution. Much like the Talmud, he also makes an interesting unintentional statement about Jesus. Celsus records Jesus as working miracles but from a dark sorcery means. When Celsus writes, he would have access to the children (who would have been elderly at his time) and grandchildren of the eyewitnesses to healing and other miracles. He clearly disagreed with the source of the power, but none the less validated inexplicable, highly improbable, extraordinary supernatural acts as historical.
The Ancient Historical Evidence
Lets put aside for a second the widely accepted historical reliability of the synoptic gospels. Papias and the Egerton fragment record miracles performed by Jesus as historical events. Even stated he knew someone first hand who was raised from the dead. Some close minded people may argue that those are bias sources. For the sake of argument, lets give them that. But Josephus, the Talmud, and Celsus are all very anti-Christian and anti-Jesus. They come from different times and Celsus from a different culture; yet all agree Jesus did inexplicable, highly improbable, extraordinary supernatural acts. They could have just as easily said, 'nope, its all a lie, he didn't do anything crazy'. But... they didn't. They would have loved to completely discredit Jesus as having anything to do with actually performing inexplicable, highly improbable, extraordinary supernatural actions; but they didn't. Anti-Christian authors, who had access to first hand witnesses and those who personally knew the witnesses, validate that Jesus did in fact conduct inexplicable, highly improbable, extraordinary supernatural actions (and to quote Josephus, "surprising feats") despite their disagreement where the power came from to do them.
Preponderance of Evidence
We must admit, just as Josephus, Talmud authors, and Celsus did; that Jesus did in fact perform miracles as the synoptic gospels have recorded. We can attempt to vilify these acts as evil sorcery but the question we must ask is this: Is giving life, healing deformities and diseases, and contorting nature to validate spiritual claims that point to a merciful, forgiving, and loving God; evil sorcery? Obviously to those who will refuse to accept the existence of a merciful, forgiving, and loving God will always reject every form of sound evidence but for those who are truly seeking; this is the question they must ask. Were Jesus' miracles evil sorcery to validate a lunatic or holy acts of God to validate his Son?