달걀국 and 계란국 are the same things. Egg soup. And you're right the ㄱin 국 is pronounced like ㄲ, because it's a compound noun. (계란+국). So, it's pronounced 달걀-꾹. or 계란-꾹.

계란 = 달걀 an egg 국 = soup 계란국=달걀국 soup made of eggs