Henny Penny is a character in a classic children's story. She is a chicken who, when an acorn falls on her head, believes that the sky is falling and spreads alarm all through the barnyard. This reminds me of the anime Chicken Little.
When someone is referred to as a Henny Penny, it usually means someone who is an alarmist, or someone who reacts to a situation without knowing all the details.
Now then I know. Sounds like hyper Gan Jiong Spider or Kan Cheong Spider (a Singaporean colloquialism) - a term used to describe people who have no patience, really anxious and eager to get work done, with a mix of Kiasunism (literally means afraid of losing in Hokkien) and chaotic panic. Similar but not the same.
Go here if you're intererested to learn more about Singlish : The Coxford Singlish Dictionary