Did the Buddha ever mention or encounter Jews? I understand that Siddhartha Gautama Buddha lived at the same time as the ancient Israelites (Jews) and that India knew of Israel and that ancient Israel knew of India and that the two nations traded with one another. I am also aware that at one time there were a small group of Buddhist staying in Jerusalem.

My question is, did Gautama Buddha ever mention Jews or did he ever encounter them? Or was he completely ignorant of their existence and nation all together?
Well, the "ancient Israelites" lived quite a long time compared with the lifespan of a single individual like Siddharta Gotama, the Buddha. Their own history, taken from the religious tradition, was from Abraham up to 0 BCE (or today, don't know when "ancient" ended...), and this means at least something from 2000 BCE; Moses is estimated to 1200 BCE if I recall right. But we (and also the Jews themselves) have this all only in the form of myths. Back to 1000 BCE and earlier I don't know whether there are any other reliable sources (besides the Torah) with such specific stories like that Torah ones about the kings Saul, David, Salomo, Rehabeam, ... and which were the first kings which were also mentioned by other tribes or nations (for instance Babylonian, Persian, Egypt) in governmental letters and statements.
From the Buddha we have only discourses (from one lifespan of about 40 years of teaching)  with Brahmins as foreign religions and some remarks about contemporary ascets, like Mahavira (I think the patriarch of the Jains - which have also survived as a religious movement up to today), the "naked ascetics" and the "white robed ascetics", the followers of Kassapa from Uruvela and so on. As far as I have read in the Pali-canon (the corpus of german translations) there was no explicite remark about other, more remote tribe/gentile/religion (also not about hellenistic ideas of the same historic times - but which in my view does not say that he might have or didn't have heard about it. In the contrary: why should you involve your disciples in a discussion about greek philosophy -even if you'd heard about it- when every important ingredient of it you can discuss with local examples...)