Like Robert, I have never heard anyone say 'He laughed his heart out'. It sounds like a translation from another language, or a something a foreign learner would say by analogy with 'He cried his eyes out'. When I googled it, all I found was one source, an 'inspirational' rhyme beginning 'Laugh your heart out, Dance in the rain'. That's all. 'Laugh one's head off' is a common idiom. 'Laugh one's heart out' is not an expression which any native speaker would use.

I'm American/Australian, and I've never heard of "he laughed his heart out." So to me, the other one is the only choice.

In my opinion they are both correct. These expressions are very similar.

I havent heard "laugh my heart out" i think he better phrase would be: "he laughed to his heart's content" "He laughed his head off" is correct a other way to that is: "he was cracking up!"