Of course, predicting future stability is complete guesswork!
The oldest country (not government) is likely Vietnam (2897 BCE). Other contenders: Japan (660 BCE), China (221 BCE), Ethiopia (~800 BCE), or Iran (678 BCE).
Few of today's modern states pre-date the 19th Century, many antedate World War II or the great de-colonialisation of the 1960s including much of Africa and Oceana (some of the longest inhabited regions of Earth).
Among the more long-lived institutions are the Catholic Church (traditionally founded by Jesus ~30 AD, emerging as an institutional power in 2nd Century Rome). The oldest company I can find is Kongo Gumi, founded in 578, a Japanese construction firm. The record however is likely held by the Shishi Middle School founded in China between 143 and 141 BCE.
My own suggestion would be the Krell, though some might disqualify this based on a requirement for human organization.
I would say a stable country is one which has had a legit democracy for 70 or so years and doesn't share a border with a non-democratic / non-legit-democratic state. These two points suggests its unlikely to have a revolution or be invaded any time soon.
Of course, if you look at the UK governments track record with IT.. you wouldn't trust it. I would say the same with the US, especially in concern with data security.
[NB I'm not English]
Japan (660 BCE) and China (221 BCE) have both had feudal / bureaucratic governments exhibiting very high levels of stability. While dynasties and eras are marked, the overall states persisted largely intact.
