I would think that for US forces in Europe, they would be unlikely to know much about the MilGov/CivGov split because the flow of information would be strictly controlled by the military heirachy. Those personnel in HQ and signals units that might hear about it would probably filter the information to support their own beliefs, meaning that even if higher HQ wanted the news to get out, some junior officer/NCO/OR might not pass it on because they disagree with it. Or the reverse could be the situation, HQ doesn't want it to get out but somebody believes the troops should know.
But I also think that once they return to the USA they would learn about it. Maybe not quickly, but sooner or later they would hear talk or even be direcly approached by people wishing to know where they stand on the issue or directly courting them to one side or another.
I can imagine both sides playing the 'political campaign' game to woo people to their side and using various methods to let people know who to throw their support behind. It may not be 'representatives' travelling the campaign trail but all the other methods of advertising their positions/claims etc. etc. would be used.
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