How would you translate "hard" in "squeeze my hand as hard as you can"?
The adverb/adjective you're looking for is '紧' (tight), which is commonly used for squeezing, seizing or grabbing something especially hands.

'Squeeze my hand as hard as you can' can be translated into '抓住我的手，越紧越好' or '尽可能紧地抓住我的手'.