I wonder what it means myself. Looking up "fantan" online the main thing that comes up is a Chinese card game but I don't see the connection. If I had to guess I would say that it is similar to "hotshot", another Hud nickname for Lon. But I can't say exactly where it comes from.
"I don't want any Commies in my car. No Christians, either."
It could mean "square." Fan-tan the game uses a square space in the center of a table. Remmeber, this was the time of the "hip" vs. the "square."
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"Why do people always laugh in the wrong places?"
--Norman Mailer, "The Hip & The Square"
I'm sure you'll never see this reply (almost 6 years after asking) and I don't recall the instance in the movie, but given that they are cowboys it could be short for "fanny tan" which may be some form of skin condition related to riding.
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