Actually, that is a very good question.
There is the belief that Job existed after the flood (cf. Job 22:15-16; 28:19; but before the law (cf. Gen. 46:13)).
Now whether it is the same Job, we can't say for sure, but there is a Job mentioned in the descendants of Issachar.
"And the sons of Issachar; Tola, and Phuvah, and Job, and Shimron."
Job, appears to be a descendant of Jacob and Rachel.
Genesis 46:15 starts the line of descendants from Jacob and Leah.
Just making a guess here, I would say that yes, Job was a "Hebrew" therefore, he was Jewish.
God Bless
Till all are one.