Why do the characters in Akatsuki no Yona have Korean names? When I watch the anime and hear the names of the characters (Soo-won, Son Hak), I thought that the manga is written by a Korean and published in Japan - like in the case of Freezing.

I couldn't have been more wrong to assume that. When I check Akatsuki no Yona on mangaupdates, it was written by Kusanagi Mizuho, whose profile suggests that he is a native Japanese.

Foreign names are common in Japanese manga. It's very common to introduce a foreigner (main character or not) to join the rest of the Japanese cast. Apart from those cases, we have stories where the setting is in a land foreign to Japan, or in an alternate universe where the names of the character are made foreign like in the case of Akatsuki no Yona. However, in most of the manga I have read that falls into this category, the names are dominantly Western and I have never seen any other shows with all the characters having Korean names.

Why does the author go to the trouble to use Korean names for the characters? Why not Japanese?
The setting for Akatsuki no Yona is loosely based on the Three Kingdoms of Korea period. The Kouka Kingdom is inspired by the Goguryeo Kingdom. Its neighbor Sei and Xing are based on Baekje and Silla respectively. You can also see that the three kingdoms are located in a peninsula in the manga and that they almost have the same layout as their real-life counterparts.