He'd better --- (not eat)anything for twenty four hours.

就填not eat.had better后面接动词原形,如果表达否定意思,直接在动词原形前面加not.
如还有不明白的,请继续追问!
再问： He'd是什么的缩写？
再答： 是he had 的缩写。he had better是“他最好……“的意思。