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...)
You could run a search on the CSCD Pali to check me (but what word would you use?), but as far as I can recall there is no mention of Jews by Gotama or anyone else in the Suttas.
There are some who believe that Christ actually studied Buddhism with the Tibetans, but this is a controversial position.
See: http://www.sacred-texts.com/bud/index.htm#modern scroll down to the bottom of the page to find titles on this subject. (A great resource, by the way, for all matters scared.)
Answering as a practicing Hasidic Jew:
I'm not sure if the Buddha himself encountered Jews, but there is an interesting anecdote from Jewish commentary on the Torah, which suggests that Abraham sent his sons -- his children from Hagar, who were not "Jewish" -- "East" bearing "Gifts." Jewish mystical readings of the Torah interpret these gifts as spiritual gifts, specifically gifts to free the East from a tendency to worship idols.
Translation of the relevant commentary here: http://www.chabad.org/kabbalah/article_cdo/aid/380365/jewish/Abrahams-Presents-to-the-East.htm
Whether or not you choose to view this commentary literally, it is, at least personally, interesting to consider how many surprising parallels I've found between Buddhist and Jewish spiritual concepts.
Simple answer is to your Question is NO
Its likely and its likely India was familiar with contemporary judaism due to Abraham's sons settling in the area and the later dispersion of the lost tribes, they had to go somewhere? Buddha was a truth seeker and a reformer and disturbed by the suffering of people in his kingdom, he forsook all and set out to find answers Israel was well known as news of the exodus spread wide and all the wilderness happenings, then the news of Babylonian captivity Nebuchadnezzar the statue, then the lions den, at that time many would have made the pilgrimage seeking truth ans wisdom, eg, Queen Sheba sought Solomans wisdom. He forsook the ascetics of his land in search of truth and definitely Jerusalem would have attracted his enquirey like a magnet attracts steel
I can't imagine why the Buddha would mention Judaism, even if he knew of it. From a Buddhist perspective, the old Judaism of Moses is a religion based on morals & petty observances, which was not dissimilar to Brahmanism. Based on the Pali scriptures, Brahmanism does not appear as 'polytheistic' as the more recent Hinduism. Although other gods are mentioned, such as Indra or Prajapati, Brahma (the Creator & Father of All) is the primary god; thus close to monotheistic.
While Jews (Isaiah 49) have self-proclaimed themselves 'A Light to the Nations', Buddhism would offer little merit to this Israelite propaganda, written in a tribal scripture (Isaiah) that prophesies doom & destruction to most neighboring nations in the region.
Being merely about morals, Judaism would have been held as similar to many other mundane worldly religions of the Buddha's time; something not offering a path of spiritual liberation. Judaism would have been the kind of socially orientated religion that the homeless ascetics such as the Buddha were running away from.
A quick look at Wikipedia confirms suspicions that some of the more philosophical or eclectic books of the Old Testament are post-Buddhist, such as Ecclesiastes, Job & Jonah. Ecclesiastes is a weak attempt at Buddhist-like wisdom; Jonah speaks of the universal compassion of God (contrary to the previous tribal focus); and, in Job, Satan appears for the 1st time, behaving towards Job similar to Mara towards the Buddha. I would speculate these books, similar to Christianity, were probably influenced by Buddhism.
While Christianity is very close to Buddhism in many of its themes & terminology, such as a 'liberated kingdom not of this world', the very fact that Christianity is not as advanced as Buddhism yet declares itself more spiritually advanced than the ancient worldly moralistic ritualistic Judaism shows how philosophically distant the ancient nationalistic tribal Judaism would have been from Buddhism.
It is considers that "sunaparantha" is somewere in midlle east. following quats from pali thipitaka,
Saṃyutta Nikāya 35 -Connected Discourses on the Six Sense Bases - 88. Puṇṇa
“Now that you have received this brief exhortation from me, Puṇṇa, in which country will you dwell?”
“There is, venerable sir, a country named Sunaparanta. I will dwell there.”
Punna, I have advised you in short. Now in which state will you abide?”
“Venerable sir, now that I'm advised in short, I will abide in the Sunaparanta state.”
