The flight data recorder will give information about the co-ordinates from where the missile was fired. Then it will be easier to ascertain who fired the missile (according to the latest regional control map).
This is com air. They do not come equipped with any sort of radar warning receiver. They cannot detect anti air missile launches. Some airlines have looked into countermeasures against IR based man portable air defense systems but such a system would be useless against a radio guided SAM. The best the black boxes will show is a loss of hydraulic pressure caused by the missile explosion and the subsequent futile efforts of the flight crew to maintain control for however long the plain remained intact.