In the last few months, my anecdotal experience is that I have had various HH properties booked at the gov/mil rate including in San Jose, CR; Panama City, Pan.; PHL; Fort Worth; Grapevine, Tex.; Houston/NASA; Merida, Yuc.; Chattanooga & Knoxville, Tenn.; Vicksburg, Miss.; Guam; Albany, NY; St. Pete., Fla.; Colorado Springs; Austin;and Southwest Virginia and only one asked for a government ID, DT Chattanooga Hampton. I stayed at others but the special rates were lower than the government so I am not sure if they would ask. 3-5 years ago I remember them asking more often for IDs.

As to the question about Canadian military, I am not sure. Sometimes my booking says "government rate" and sometimes "US government rate". I worked with many Canadian officers on NORAD missions and they often would ask if they could partake in the military discounts at businesses, and when they travelled with us as crew they got the same hotel rates (and UA even let them pre-board as they would just say uniformed military personnel instead of US Military Personnel).

I would say go for it, what are allies for ;-)