What is Ponyboy's issue with Johnny's suggestion that they disguise themselves in The Outsiders?

Ponyboy does not want to cut and bleach his hair because it is fundamental to his identity as a greaser. 

As Ponyboy tells us repeatedly, hair is very important to greasers.  Unlike the Socs, the greasers do not have much money.  They distinguish themselves through their hair.  It doesn’t really take money to grow your hair out and style it, and so this is what the greasers do.  

When Pony and Johnny are on...

Ponyboy does not want to cut and bleach his hair because it is fundamental to his identity as a greaser. 

As Ponyboy tells us repeatedly, hair is very important to greasers.  Unlike the Socs, the greasers do not have much money.  They distinguish themselves through their hair.  It doesn’t really take money to grow your hair out and style it, and so this is what the greasers do.  

When Pony and Johnny are on the run, they hide in an old church outside of town.  Johnny goes out to get supplies and returns with a deck of cards and peroxide.  Pony gets worried as soon as he sees it.  Johnny tells him that they have to disguise themselves so they don’t fit the descriptions in the paper. 

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. (Ch. 5) 

Johnny points out that if they get arrested their hair will be cut anyway.  Johnny explains that it is a way to break them, because “they can't take anything away from them because they don't have anything in the first place.”  The only thing they have is their hair. 

In fact, when Pony returns home the other boys tease him about his hair.  Two-Bit points it out as soon as he sees him, to Pony’s dismay. 

"Man, dig baldy here!" He was staring at my head as he circled me. "I wouldn't have believed it. I thought all the wild Indians in Oklahoma had been tamed. What little squaw's got that tuff-lookin' mop of yours, Ponyboy?" (Ch. 7) 

Losing his hair is part of the transition Pony undergoes from the time they go into hiding.  After the fire, Pony, Johnny, and Dally are all injured.  Johnny’s injuries are life-threatening.  Nothing is the same for Ponyboy ever again, and he questions his very identity as a greaser.  As Johnny points out, Ponyboy isn't really the rumble type.  He has a chance to make something of himself.