Why did both Northerners and Southerners reject John Crittenden's compromise?

For the most part, I would say that this question misstates the facts about the “Crittenden Compromise.”  As you can see in the link below, when the compromise came up for a vote in the Senate, all of the senators who voted against it were Republicans.  This means that they were all from the North as there were no Republican senators from the South in those days.

Northerners opposed the Crittenden Compromise because they felt that it gave too much to the South.  Free soil advocates like Abraham Lincoln rejected the idea of allowing slavery to expand farther than it already had.  They did not like Crittenden’s ideas because his compromise would have guaranteed that slavery would exist everywhere below the Missouri Compromise line that was not already a state.  This meant that slavery would exist outside of the areas where it was legal at the time.  This bothered many Northerners, and it would have required the Republicans to give in on most of their major positions with respect to slavery.

If we have to say that the South opposed the compromise, we can say that they did so because they just wanted to secede.  There were six senators who abstained from voting and all of them were from Southern states that were in the process of leaving the Union.  We can say that they opposed it because they just wanted to create their own country instead of trying to compromise with the North.