I speak Korean only marginally, so I invite someone truly bilingual in that and English to correct me, but that said...
han-guk is the romanisation of the syllables that in Korean are one way to say Korean as in han-guk-saram='Korean person' and mee-guk-saram="American person'.
In these examples, the particle guk (see also the writeup for gook) means person/people. Now the interesting thing is that in my beer-clouded memory (I drank a lot when I lived there), the particle han can also mean sad. Given Korea's long sad history of having the shit kicked out of it over and over again by the Japanese and Chinese, that's appropriate.
The other thing that twigs is the use of the word word-sound 'Han' in conjunction with Chinese... as in the Han Chinese.