“Go to bed with the chickens” means “to go to bed very early”.

Usage in a TV series (“The Fresh Prince of Bel-Air”):
- You know, you were out so late last night, I thought you would sleep in.
- Oh, honey, I’ve been going to bed with the chickens all of my Iife. But, you know, when you’re out having a good time… the last thing you think about is the bed.
- Well, I’m glad to hear that.