Why does Shinnosuke ask people if they like peppers?
The progression of the phrase goes like this: 


  ピーマン好き (Do you like [green] peppers)？ ニンジン食べれる (Do you like to eat carrots)？ 納豆にはネギ入れる方？ (The method of sticking negi [spring onion] into natto [fermented soybeans] "Are you the type who puts spring onions in their natto?")?


This phrase doesn't translate very well, but it's an equivalent to South Park "fish sticks" gag: 


  Do you like to put fish sticks in your mouth?