It’s Jay-pan fan, and it is referring to the folded Japanese fans as shown in terntii’s link, but not necessarily that elaborate.
The purpose of the fan is to flirt, think of all the old movies where a young woman peeks coquettishly over her fan and hides her giggles. June is saying she’ll be dancing, having a good old time and flirting with the boys.
Snowball might be better understood through the phrase “take it by storm”, which the dictionary defines as “have great and rapid success in a particular place or with a particular group of people”. A snowball’s nature is to collect as it goes. As the song indicates that the man’s intention is to be adored like a King, “People gonna stoop and bow” and connect with many women “All the women are gonna make me teach 'em” the snowball reference is put into context.
It’s important to understand a text under the cultural values of the time that it is written. The connotative vulgarity associated with “snowball” (verb) in 2023 would never be sung on a public stage or put on the radio 1963. Homosexuality was still illegal in the commonwealth and most of the USA. The general population was conservative; Elvis’s pelvic movements while dancing were considered vulgar at the time.
I was under the impression that she would be laughing at him from behind the fan. He was going to get run out of town with his tail between his legs as she laughed at him. I’m not old enough for the song but I only seem to see those fans in media when someone was laughing/giggling from behind them.
Welcome to the Straight Dope, @Analysis_101! You might want to note that you’re responding to a thread that started 18 years ago and was revived in 2013. Most of the original posters are no longer around.
Further, I doubt “snowball” had the same slang meaning in 1963 that it does now. However, I do think there’s a bit of sexual innuendo in the song, specifically when June sings “They’ll lead you round town like a scalded hound with yo’ tail up between yo’ legs”.