Yes, they lost the war and were conquered militarily so they could not leave the Union. Therefore they had to rejoin. But before they could regain the full powers of a member state in the union, congress required that the ratify the 14th Amendment to the constitution, guarantee the vote to black males, and prevent former high Confederate officials from serving in state government.
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Due to the stubborn resistance opposed by the former Confederate States (with exception of Tennessee) to Reconstruction and their refusal to ratify the 14th Amendment, the Congress passed in March 1867 a Reconstruction Act that imposed its version of a postwar settlement by means of a diktat.
The States in question, in groups of two, were organized into five military districts each one under the rule of a military governor having extensive powers. Then after restoring law and order, the States had to organise conventions to amend their own constitutions so as to conform with the Constitution of the United States.
After completing these stages the conforming States could be readmitted in the Union.
More rights for former slaves
The southern states threatened to leave the Union because of antislavery talks.
An agreement to end slavery immediately
The North was called the Union because it represented those states that remained in "union" together as the Unites States, and because it was fighting to preserve that union by keeping the rebellious Southern states from leaving it.
When Lincoln took office in March 1861 seven southern states had left the Union.