If we look at the character 国 gúo as a graphical presentation, we can find the compounds "jade", 玉 yú , and "encircle", 囗 wéi, in it.

This character is a so-called simplified character, and the compound meaning jade was added into it after simplified characters were implemented in Chinese Mainland around the middle of the 20th century. Maybe adding the jade into this word describing "country, nation, state" puts some additional glory on the concept, or maybe it symbolizes the ruler and his power.

Earlier inside the 囗 wéi-part there was a compound that stands for "area, territory", hùo 或, inside which we can find 戈 gē, a kind of weapon. A state is an encircled area capable of defending itself.

China is called Zhōnggúo 中 国 in Chinese. It is mostly translated in English as "The Middle Kingdom". The character 中 zhōng very likely originates from a flag, that stood as a symbol for the power of the emperor. Another often repeated interpretation for this character is that it simply depicts an arrow that hits the middle (zhōng) of its target.