I was just back from ICT, and the MR personnel in my unit got a Swiss Army quartz watch. I didn't get to see the case back, but the dial also has some logo representing the SAF.
Back to the Hamilton, I understand that there are procedures that essentially file off the engraving. You sorta lose some metal this way.
Just in case you are thinking of selling the watch, don't expect the watch to fetch too much. A regular BNIB automatic Hamilton watch can be had for more than 40% off the retail price. (Standard discount should vary from 20-30%) Given the fact that people would know this is a free gift from the military that most Singaporean males will receive eventually anyway, I won't put too much hopes into that.