It could be either box or dish. The disk has amplifiers in in the receiver "horn". They are powered from your box. The dish amplifiers could have even been damaged from the proximity of the lightning. I have even seen coax that was damaged by a lightning strike.
If one of your neighbors has the same satellite service, maybe you two can swap a boxes around to troubleshoot your outage. There are satellite downlink signal strength meters that could help troubleshoot.