In Spanish, the pronunciation of the Y doesn't depend on the area. It is always pronounced [j] (Y as in "yes").
If they were pronouncing a Spanish name beginning with Y- with the English J-, they were simply pronouncing that not in Spanish but in English. Since the sound of the English J- doesn't exist in Spanish, the Spanish speakers adapt it as Y-, which is the more similar in their language (even being very different), and that is why some English names with J- as Jessica, Jennifer or Jonathan has been adapted in Spanish as Yésica, Yénifer and Yónatan.
Very probably, many of those Mexican immigrants are not more Spanish speakers but of Spanglish (a mix of Spanish and English) or directly speakers of English with Spanish influence.