She remembered Ezekial and thought: He walked north for freedom and I am walking south.
The previous quote can be found early in the chapter. Lyddie has decided that she is going to head south and try and get a job at the mills. A customer at the tavern told her about the opportunities that are there for hardworking young girls. Lyddie doesn't immediately search out that opportunity, but the issue is forced in chapter 6 when she is fired from her tavern job.
In order to get to the mills, Lyddie has to head south. She is excited about the opportunities that she might find there. She believes that she will be working for herself and earning solid pay for it. She will not be an indentured servant, and that sounds like freedom to Lyddie.
Journeying south for this is ironic in a couple of ways. Generally speaking, the South is associated with slavery in America. Runaway slaves would head north for their freedom, so it is odd for a reader to hear freedom and the South linked together like that.
If you know the entire story, then the chapter title is ironic for another reason. Once at the mills, Lyddie becomes a slave of the mills and the machines. She is a piece of the factory machinery that can be hired, used, fired, replaced, and so on in the same way that any other machine could be. She isn't treated as a valuable person. She is a slave, and Lyddie tries to deny it over and over again, despite other girls in the house repeatedly making the comparison between their work and slavery.