Ireland is actually fairly diverse from an ethnic standpoint. Like England, there were Viking raids that left a Norse influence, and in fact Dublin was founded by Vikings. The area around Dublin was settled by English settlers who eventually "Gaelicized" and became Irish, Northern Ireland was also settled by English/Scottish settlers in an attempt to pacify the region. So many Irish have Norman/Germanic/Norse influence as well as their Celtic and Proto-Celtic roots.

That being said, your question in regards to the Basques goes much further back than all that. It's difficult to trace each nation's exact ethnic ancestors, since ethnicities are formed through interactions with other ethnicities.(as I mentioned above)

EDIT: Listen to /u/matts2 's post about how 'Celtic' doesn't have much to do with genetics.

Also, ethnicity is usually defined as being how the group labels itself. Irish people certainly identify as Celtic, whereas Basque people do not. From a historical vantage, this is more important than genealogical lineage.

This is really more of an anthropological question to be honest, you could try /r/anthropology if you're interested.

Sources: for Irish origins

for more modern Irish history, my source is an Irish studies class I took last year.

Genetically "Celtic" does not seem to mean anything. This shows a map of the genetic distinction across the Isles. The largest contribution was post Ice Age, pre-Celtic culture.

I get the impression that the scholarly consensus is the latter. There is an interesting page on this by Simon James of Leicester University here http://www.le.ac.uk/ar/stj/alternative.htm.