I did answer a very similar question very recently: Why did Northern Ireland stay with the UK, while the rest of Ireland became independent? What made the people of N. Ireland be so much more supportive of the UK? Did N. Ireland have more British immigrants in it and that's why? If so, how did it come to that?

Of course, theres a lot more that can be said. The framing of the question does suggest a bit of confusion on your part, ie. When Ireland gained its independence why did Northern Ireland not jump at the chance? In reality Northern Ireland was a product of the process of Irish independence and partition. It did not exist before and only came into existence as a direct consequence of Irish independence.

To add something a bit more on what I said in that previous post, you note the fact that the large Protestant population gives NI closer ties to Britain but wonder why they didn't jump at the chance. Ultimately that was the bottom line, Ulster Unionists feel that are British and the idea of separation from the United Kingdom was (and continues to be) a source of deep ideological anxiety. They felt that they were as a British as someone from England and the idea of independence was unthinkable.

Of course, not everyone in what became Northern Ireland thought this way. There were plenty of Irish nationalists too and some areas of the country which had Catholic nationalist majorities (e.g. Derry) where included in the partition anyway. But the Ulster Unionists, along with British Conservatives, had sufficient influence to shape the outcome that they wanted. A boundary commission was set up which, it was believed, would more accurately mark out the border but in the end it changed very little.

The Britishness of Ulster unionists has its roots in the settler colonialism of the Early Modern period as noted in the previous post. On its own it provided a potent enough reason against independence. Of course, there were other factors which shaped this ideology though. Firstly, their deeply held Protestantism (often rather radical forms of providential Protestantism too) was in stark contrast to the majority Catholicism of Ireland was a whole. It was feared among certain groups of unionists that there religion would be under threat in an independent Ireland. A popular slogan of the day was that "Home Rule would be Rome Rule".

We should also consider the material benefits that parts of NI enjoyed at the time. In stark contrast to the economic drain it exerts today, the area of north east Ulster in the 19th and early 20th century was an industrial powerhouse within the United Kingdom. The shipyards of Belfast where one of the key economic centres. This also marked a strong contrast to the rest of Ireland, which was generally speaking much more rural. At the time Northern Ireland, or parts of it at least, enjoyed a number of benefits as part of the United Kingdom and, indeed, the British Empire.

Just typing this from my phone, so if you want any elaboration or specific references for anything then let me know. In the comments to that previous question I provided some suggestions for further reading, but if you want something more specific let me know.