I only thought 국 was used in relation to countries. Literal translation would be 'medicine country' if I had to guess.
국 corresponds to following hanjas: 國, 菊, 局, 鞠, 鞫, 麴, 匊, 掬, 跼, 麯
Moreover 국 is a native word for 'soup'. As you can see, 국 has a ton of meanings, and country (國) is only one of them.
The one used in 약국 is '局' and, according to wiktionary, it means 'office', 'shop' or 'department'. Makes sense to me.