Naturally, countries are most resilient than your average human, thus, the possibilities of dying are slightly different for them. They can still die from both blood loss, and wounds, although these wounds have to be directly inflicted by another country, since a human alone cannot kill a country.
Although uncommon, countryhumans can also commit self-death. This has only happened twice, once with Ptolemaic Egypt and then later on Third Reich.
Finally, countryhumans can also die through what is called "drifting away" or in technical terms, the dissolution of the country. It all begins with several headaches, then migraines, then fainting/passing out frequently, up until the country collapsed and it's body drifts away. Through the entire process, small cracks will appear on the country's skin, worsening depending on what stage of the process a country is.
(To clarify, country's can suffer from sicknesses and such, but it can't directly kill them.)