Josephus also mentions someone named Saul, who lived at the same time as the other people you mention.Acts 13:1 says: "“Now in the church at Antioch there were prophets and teachers: Barnabas, Simeon who was called Niger, Lucius of Cyrene, Manaen a member of the court of Herod the ruler, and Saul.”
In this one sentence, there are three, or possibly even four, names very similar to names associated with the Jewish war
Ant. 20.9.4: "Costobarus also, and Saulus, did themselves get together a multitude of wicked wretches, and this because they were of the royal family; and so they obtained favor among them, because of their kindred to Agrippa; but still they used violence with the people, and were very ready to plunder those that were weaker than themselves. And from that time it principally came to pass that our city was greatly disordered, and that all things grew worse and worse among us."
War 2.17.2: "So the men of power perceiving that the sedition was too hard for them to subdue, and that the danger which would arise from the Romans would come upon them first of all, endeavored to save themselves, and sent ambassadors, some to Florus, the chief of which was Simon the son of Ananias; and others to Agrippa, among whom the most eminent were Saul, and Antipas, and Costobarus, who were of the king's kindred; and they desired of them both that they would come with an army to the city, and cut off the seditious before it should be too hard to be subdued."
And in War 2.20.1 he is mentioned along with his friend named Philip (another name found in the NT):
"After this calamity had befallen Cestius, many of the most eminent of the Jews swam away from the city, as from a ship when it was going to sink; Costobarus, therefore, and Saul, who were brethren, together with Philip, the son of Jacimus, who was the commander of king Agrippa's forces, ran away from the city, and went to Cestius. But then how Antipas, who had been besieged with them in the king's palace, but would not fly away with them, was afterward slain by the seditious, we shall relate hereafter. However, Cestius sent Saul and his friends, at their own desire, to Achaia, to Nero, to inform him of the great distress they were in, and to lay the blame of their kindling the war upon Florus, as hoping to alleviate his own danger, by provoking his indignation against Florus."