Why does Pony have a problem with Johnny's idea to disguise themselves?

From the text:

"Oh, no!" My hand flew to my hair. "No, Johnny, not my hair!"
It was my pride. It was long and silky, just like Soda's, only a little redder. Our hair was tuff--- we didn't have to use much grease on it. Our hair labeled us greasers, too-- it was our trademark. The one thing we were proud of. Maybe we couldn't have Corvairs or madras shirts, but we could have hair.