If, by "country", you refer to the sovereign territory of a Nation-State, and if by "furthest North" you mean extends furthest to the North, then the answer is almost certainly Denmark, which includes the territory of Greenland. Canada (Ellesmere Island) would run a close second, followed by Russia (Franz Josef Land) and Norway (Svalbard). It is possible that there are minor islands further to the North than Greenland, which might be part of the territory of another Nation-State (most probably Canada, Russia, Norway, Finland, or Sweden), but that seems unlikely.
On the mainland, Panama is the furthest south.
However including islands, Costa Rica is the furthest south.
In the United States Maine is the farthest state north on the main land. Alaska is the farthest state north in North America.
Mexico is the farthest south north American country
Panama is the country furthest south in North America
The northernmost country in North America is geographically Denmark (via Greenland, a Danish territory).
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Panama
The country that extends farthest north is Norway, and the country that extends farthest east is Russia. Russia holds the title for being the country with the largest east-west extent.
Alaska
Alaska.
North America is the continent farthest north. Greenland, an autonomous territory of Denmark in the North Atlantic Ocean, contains the northernmost point on the continent.
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Egypt. Egypt is the country which is the farthest east in North Africa.