Definitely not the Greeks.
Which UEFA Championship - there are a number of them.
I'm guessing the Italians.
No, not the Greeks.
You are right that there are a number of championships, but there is only one in which countries play against each other, and not individual clubs (unless you count the under-21 tournament, which I don't).
Italy is a good guess, but they've only won once, in 1968. The winningest country is Germany, having won the 1972, 1980, and 1996 games. They also lost the championship game in 1976, 1992, and 2008. Next is Spain, having won the 1964 and 2008 games, and finishing second in 1984. Third is France, having won in 1984 and 2000.