Maybe not exactly what you're looking for, but Chang Apana, the inspiration for Charlie Chan, worked as a cowhand (a paniolo) in Hawaii before joining the police force.

His handiness with a bullwhip (pictured here in the Honolulu Police Museum) dates to his cowboy background. It was, ostensibly, the only sidearm he carried, and he had a pretty tough beat.

Jews have a long history in Texas dating back to the Inquisition in Spain. Here is an article briefing the history.

Another aspect are the crypto-Jews of Texas. Many Jews were forced to abandon or hide their heritage.Here for more on that.

Due to the early Jewish settling of Texas, Mexico, and the southwest, it is very likely that many cowboys and vaqueros were of Jewish descent.

edit: This is a book that might be relevant to your interests.

Well, technically not cowboys, but in that East-Meets-West-With-Swords-And-Bandits vein: Japanese samurai known as chinos guarded silver shipments in New Spain in the 17th century. Charles Mann talks about them briefly in 1493. He says that "many hundreds" of them were employed between Mexico and Peru in the early 17th century, esp. after Japan closed its borders in the 1630s (effectively exiling any expatriates). I am surprised no one in Hollywood (to my knowledge) has done a samurai Western of this specific sort yet.

Would you accept dog mushers?

There are some great answers from a previous post. Were there ever any Asian/Asian-American cowboys?

Asians were pretty limited in where and how they were allowed to work. The Chinese were famous for working as cooks and in laundries throughout the west. If they managed to make it out on the trail they would probably be working as the cook rather than as a "cowboy".

Follow-up question: does anyone have anything about Jewish Gauchos in Argentina? This question inspired me to get a couple of books off the shelf but I can't find anything beyond a single paragraph about Jewish gauchos in the context of xenophobia in Richard Slatta's Gauchos and the Vanishing Frontier. I know I've read about Jewish gauchos before but I can't think where.