In the 17th century settlers came to America from various nations of Europe, including the Netherlands (United Provinces), Germany, even Sweden and Finland who created a colony in the Delaware, which they called New Sweden.
The Swedes brought their typical wooden cabin to America and a lot of Dutch words entered AmE. The word 'Yankee' for instance (Janneke
: Little John
The Brits gradually gained control of the territories, notably New Orange
, after the Anglo-Dutch War in the Westminster Treaty (1674). The last British colony was Georgia, founded in 1732 and became a Crown colony in 1755.
After the Seven Years Wars and the Paris Treaty (1763), the Brits got the Canada and all the Territories on the East of the Mississippi. The French had until then all the territories on the West, Louisiana (much larger than present-day Louisiana) but the Paris Treaty gave Louisiana to Spain, their Bourbons allies (Choiseul obtained that, in the hope to regain it, one day, which happened in 1800 but Buonaparte was quick to sell it to Jefferson).
I don't know if there were already some Irish among these, probably so. But I guess most came after the Great Famine (1845-52).