Shouldn't the answer be the other way round? 'Saya makan apapun'?
Yes, you're right, that's also possible, but I think it sounds even better this way.
I think you also be able to say saya makan apa saja,right?
It works
Would it be more correct that translation should be "I eat anything" ?
Depends on context. "I eat anything" sounds more neutral to me than "I eat whatever" which feels more casual or carefree.