The Northern Ireland population have usually supported either membership of the UK or reunification with the Republic of Ireland. There has been very little support historically for independence. The most important reason is the weakness of the Northern Irish identity as a separate national identity. The six county region was only established in 1920 and even the idea of partition only dates from about 1913. Any move to independence would have to come from the Loyalist population and there are some fringe movements of this sort (such as the New Ulster Research Group) but most Unionists want a close connection to Britain.

There are huge economic issues with an independent country of 1.5 million people, few natural resources and dependence on British markets. It was this that killed of a movement within Unionism for Dominion status in the late 1940s caused by opposition to the welfare state.

Irish nationalists have had little interest in Northern Irish independence as they associate themselves with an Irish Republic (either the current one or a radically reformed version) and as a minority they would lose their existing protections in a majority Unionist state.

Sources

Graham Walker, A History of the Ulster Unionist Party: Protest, Pragmatism and Pessimism ( Manchester University Press, 2004 ).

It's Northern Ireland, and the demonym is Northern Irish.

There was actually a border poll in 1973, where the question of whether or not NI should remain part of the UK was taken to the people. The majority of Irish nationalists boycotted it, but a slim majority of the electorate (58%) turned out and voted overwhelmingly to remain part of the UK.

Or have more North Irish support the UK than it would seem?

I'm not sure what you mean here. The majority of people in Northern Ireland have, since its inception, wanted to remain part of the UK.