I'm a little bit late, but as the other two answers are quite vague, I thought I would give it a chance.

It depends on what you define as "Basque Country". If you're talking about the Basque Country as it is currently defined, then the answer is trickier. It is never been independent with the same borders that has now, or at least, its seven constituent territories (four in the Spanish side, three in the French side) have never been independent all at the same time, but yes at different points.

There has been two instances of more or less sovereign territories who remain as the closest historical examples we have of what could have been a Basque state. One is the well-known Kingdom of Navarre, which was fully independent from 824 to around 1620 (1512 south of the Pyrenees), but with varying borders. All of the territories that make today's Basque Country have been at some moment part of the Kingdom of Navarre, as well as some others that are not part of the Basque lands.

The Kingdom of Navarre south of the Pyrenees was slowly conquered and annexed by the Kingdom of Castile, starting with the conquer of Biscay in the year 1200 and ending with the fall of Pamplona in 1512. The Kingdom north of the Pyrenees survived until 1620, when the French monarchs (who had long before inherited the Navarrese crown) merged both kingdoms.

The other example is the Duchy of Vasconia (or Wasconia), a semi-independent duchy created as a buffer state of the Frankish kingdom with the intention of controlling the often rebellious Basque populations. It originated before the Kingdom of Navarre (the County of Pamplona was originally part of the Duchy until it secceded) and was not completely independent, but rather a vassal of the Franks. The Duchy included territories that were Basque (or Aquitanian) speaking at the time but that currently do not identify as Basque (basically modern-day Aquitania and Gascony) plus the present-day territories of the Basque Country south of the Pyrenees.

The Basque region of Spain was once the independent kingdom of Navarre.

The kingdom of Navarre (Originally Pamplona) existed from 824 to 1620. The south was annexed by Castile and Aragon in 1512 with the remainder merged with France after a period of personal union in 1620.