Jones married a jewish woman. He had kids with her who are thus half jewish.
They're fully jewish according to mainstream judaism' standards https://www.jewishvirtuallibrary.org/who-is-a-jew#halacha
Who is a Jew According to Halacha? According to Jewish law, a child born to a Jewish mother or an adult who has converted to Judaism is considered a Jew; one does not have to reaffirm their Jewishness or practice any of the laws of the Torah to be Jewish. According to Reform Judaism, a person is a Jew if they were born to either a Jewish mother or a Jewish father. Also, Reform Judaism stresses the importance of being raised Jewish; if a child is born to Jewish parents and was not raised Jewish then the child is not considered Jewish. According to the Orthodox movement, the father’s religion and whether the person practices is immaterial. No affirmation or upbringing is needed if the mother was Jewish.
Besides for differing opinions on patrilineal descent, the various streams also have different conversion practices. Conversion done under the auspices of an Orthodox rabbi, entails Jewish study, brit milah (for men), mikvah (for both men and women) and a stated commitment to follow the laws of the Torah. Conservative conversions use the same requirements as the Orthodox do; however, conversions by the Reform movement and other streams do not have the same requirements. Since the conversion practices are not uniform, many Orthodox Jews do not recognize Reform or Conservative conversions as valid and, hence, do not consider the converts Jews. Once a person has converted to Judaism, he is not referred to by any special term; he is as much a Jew as anyone born Jewish.
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