Maybe difficult to explain to non-Basques but the historical name of the Basque Country is Euskal Herria (same meaning, literally). In the late 18th century, the founder of the Basque Nationalist Party (EAJ-PNV), Sabino Arana, a very enthusiastic and catholic man with poor knowledge of the Basque language, invented a whole array of neologisms, most of which have been discarded but a few have taken some root or even become common Basque words. One of them was the word Euzkadi (sic) that he meant to mean also Basque Country, and that has been corrected by the Academy of the Basque Language to Euskadi (Arana had the weird idea that ethnonyms Euskara and its derivate root Euskal- derived from E(g)uzki: the Sun - what is very patently wrong). Whatever the case, Euskadi (sometimes still spelled by PNV nostalgics as Euzkadi) became a common word and roughly a synonym for the much more traditional Euskal Herria, but with a more political meaning: it means more like the polity of the Basques, be it sovereign state or autonomous region, while Euskal Herria means the country, the nation as such.
In the last decades the often corrupt and always compromising policies of EAJ-PNV (the major party in the Western Basque Country but a tiny force in Navarre and the North) have lead them to concieve Euskadi as only the autonomous Basque Country (the three western provinces, the area that holds officially that name in Spain), while more consequent nationalist parties, as well as ethno-realistic internationalist forces rather prefer the concept Euskal Herria normally and the idea of the Basque Country as a totality of 7 historical lands both sides of the border. In fact it's surely the most popular concept of Basque identity, maybe because it's flexible and describes am ethno-historical reality, rather than just a political project.
Now going to the grain, the case is that the Basque Football (soccer) Selection had decided last year by widespread consensus to use the term Euskal Herria. But now the PNV has decided to impose the name Euskadi instead. Players basically have been given the choice to either accept that imposition or not to play under Basque colors (there's a match against Iran this month) but the discontent is clear.
Additionally other political forces, the Nationalist Left Bloc, Eusko Alkartasuna (social-democrat nationalists) and United Left (Spanish republican and federalists) oppose the move.
The main orchestrator of this coup has been Iñigo Urkullu (yes, he uses the Spanish spelling for his name), president of EAJ-PNV, who wrote on his blog that:
As nationalist of the PNV, Euskadi is our political project. Another thing is Euskal Herria, a socio-cultural concept of the Basque People itself. But our political project for the country - for the nation that is supposed to be represented by a selection that, in this case, is born from a Basque federation of three lands with vocation of allowing players from other lands - is Euskadi.
More clear... water: the so-called Basque "Nationalist" Party is against the Basque Country and wants to create a pseudo-nation, actually just an autonomous region within Spain with the currently autonomous Western region. They don't care about Navarre nor the North: they prefer to divide the country and this is just the last epysode of a surrendering strategy. Already in he 80s, that party broke up (that's the origin of Eusko Alkartasuna) because they pacted with the post-fascists, allowing them to rule Pamplona in exchange for support in Bilbao. Since then all the policy of the PNV has been to sow corruption (and rip the benefits) and to behave as a pathetic regionalist party that would be the shame of Arana himself.
Source: article (in Spanish) at Gara.