Since you can put "parent.child" dotted notation and you don't have to put parent["child"] which is also valid and useful, I'd say both ways is technically acceptable. The parsers all should do both ways just fine. If your parser does not need quotes on keys then it's probably better not to put them (saves space). It makes sense to call them strings because that is what they are, and since the square brackets gives you the ability to use values for keys essentially it makes perfect sense not to.
In Json you can put...
>var keyName = "someKey";
>var obj = {[keyName]:"someValue"};
>obj
Object {someKey: "someValue"}
just fine without issues, if you need a value for a key and none quoted won't work, so if it doesn't, you can't, so you won't so "you don't need quotes on keys". Even if it's right to say they are technically strings. Logic and usage argue otherwise. Nor does it officially output Object {"someKey": "someValue"} for obj in our example run from the console of any browser.