Why did southerners support slavery if they didn't own slaves?
While it is true that not all southerners supported slavery, it also is true that not all non-slave owners in the south supported the end of it. In fact, there were many southerners who greatly feared what would happen to the South if slavery were abolished.
To best understand this thinking, put yourself in the shoes of a southerner around the time of the Civil War. Many southerners—including those who didn't own slaves—felt that Abraham Lincoln was trying to minimize the role the southern states played in the United States. If they didn't fight alongside their friends and neighbors who owned slaves, many believed that they would have no chance to be accurately represented in Washington, D.C.
Those who defended slavery—even if they didn't own slaves themselves—believed that the slaves in the southern states were better taken care of than the poor who were found in the northern states as well as throughout Europe. Some argued that by keeping blacks as slaves, slave owners were protecting them by providing for them homes in which to live and healthcare when they were sick.
Southerners also feared what would happen if so many slaves were freed in such a short period of time. They believed widespread unemployment and chaos would be imminent once slaves were freed. By continuing the status quo, the order and supposed harmony of society as it existed would remain. For many southerners, this was desirable—even if they didn't own slaves themselves.
I would also like to add that many Southerners who were not slave owners still had a financial stake in the continuation of slavery. The South was based upon an agrarian, slave-holding economy. If there were no slaves, there might not be plantations, and while slaves did a great deal of the work, there were paid jobs for white people on on plantations. Ships would not need to be built or manned if there were no slave trade. Slave bounty hunters would have no work. Stores would not be able to sell their goods if plantations were not producing. Middlemen would have no goods to move along. White Southerners might very well find themselves competing with former slaves for jobs.
The other aspect of this that I think is worthy of note is that for many people, having a class of people they are able to consider beneath them is what props up their self-esteem. People who were plantation owners were the upper crust of Southern society, and the people who were not slave owners were likely to have felt that having enslaved African-Americans around is what kept them from being at the bottom of the hierarchy. And in fact, every effort was made by Southerners to keep African-American at the bottom even once the slaves were freed.