Your questions go to the central arguments of the conflict. While the lawmakers and politicians in the states that seceded (the Confederacy) did indeed feel that they lived in a nation-state that allowed them to break off, the north (Union) held firm to the notion that all states within the United States of America were part of one country. The Union saw any attempt to break free as an act of rebellion, and thus raised an army to put the insurrection down (just as Canada might if British Columbia decided to secede).

So getting back to your question, I guess we have to define the "they." There certainly was a "they" that believed "ending" the Union and splitting off was a justified action, and reasonable. They were motivated by the question of slavery, and a long-standing debate in the U.S. regarding small vs. big government. On the other hand, there was a "they" that believed the United States was just that: a united collection of states that formed a country which could/should not be broken up and divided. A very bloody civil war followed to settle the question, and over time, it evolved into a struggle over emancipation as well (though from the start of the war, this issue was front-and-center).

What ultimately kept the country together was the bravery and dedication of the soldiers fighting for the Union, and Abraham Lincoln. Through canny political maneuvering, determined persistence, intuitive strategizing, and a willingness to make mistakes and learn from them, Lincoln successfully prosecuted the war under conditions that would have broken or beaten other individuals. He is a great example of a person whose entire life built up to this one moment: where all his political experience, self-taught learning, acquired people-skills, and learned-ability to overcome crushing defeat made him the man that was needed to face a monumental challenge. Lincoln also surrounded himself with the very best people (Seward, Stanton, Grant, etc.) and trusted in their ability to do their jobs.

[Sources: Bruce Catton, 'Mr. Lincoln's Army,' & 'Grant Moves South'; Doris Kearns Goodwin, 'Team of Rivals'; Jay Winik, 'April, 1865']