If he had amnesia, how'd the guy who found him know his name was Gregor?
It's probably printed on Gregor's armor and equipment (which the diner owner was keeping stashed away) so it wouldn't get mixed up with other Clonetrooper's gear back when he was still in the Army.
This episode was an improvement over the previous dismal droid/Gascon episodes, if only because Gregor is a compelling character, but the whole "You took me away from the slave army I was serving in and turned me into a slave dishwasher!" thing was bothersome. Not from Gregor's perspective, since obviously the Clone's loyalty to the Republic and later Empire has been programmed into them so they wouldn't question their role (with a few exceptions), but from the series' and franchise's perspective, where the nature of the clones as slaves and the Jedi's/Republic's hypocrisy on that point has continually been glossed over. If they're just making that statement from Gregor's perspective then it's not as big an issue, but if the showrunner's really think there is a distinction between the two, that would bug me.