If the Holy Spirit never physically indwelled anyone in the Scripture; how is one to explain the concept of "the Spirit came upon them?"
I don't believe I ever said that the Holy Spirit never physically came upon anyone, though the Spirit is, well, spiritual and not physical. When Jesus was baptized we are told, "He saw the Spirit of God descending like a dove and alighting upon Him" (Matthew 3:16). And when the Apostles received the Spirit, "Then there appeared to them divided tongues, as of fire, and one sat upon each of them. And they were all filled with the Holy Spirit and began to speak with other tongues, as the Spirit gave them utterance" (Acts 2:3-4).
Yet the verses used to argue that the Spirit comes upon people today are not saying what is being ascribed to them. There are promises that the Spirit will dwell with the Christian. Such refers to a close association and is the same idea that God and Christ dwell with the Christian. The phrase "the Spirit came upon ..." refers to direct guidance by the Spirit, such as Moses and the elders of Israel (Numbers 11:25), or judges of Israel, such as Gideon, (Judges 6:34). Often the coming of the Spirit was accompanied by the ability to prophesy or to do mighty works.
For more details see: Indwelling