You should have stuck with your first instinct, because it isn't true. Even the quote your source used fails to support the assertion. "In the absence of negro slavery there must be white slavery" does not imply "In the presence of negro slavery there should be white slavery." It becomes even clearer when you look into his works, rather than what someone else thinks they said. In the 1856
"Sociology for the South, or, The Failure of Free Society", which has come up on here before, he argued:
There's also
"Cannibals All! Or, Slaves Without Masters" from 1857, where he argues:
"In slave society, one white man does not lord it over another; for all are equal in privilege, if not in wealth; and the poorest would not become a menial - hold your horse and then extend his hand or his hat for a gratuity, were you to proffer him the wealth of the Indies. The menial, the exposed and laborious, and the disgraceful occupations, are all filled by slaves. But filled they must be by some one, and in free society, half of its members are employed in occupations that are not considered or treated as respectable. Our slaves till the land, do the coarse and hard labor on our roads and canals, sweep our streets, cook our food, brush our boots, wait on our tables, hold our horses, do all hard work, and fill all menial offices. Your freemen at the North do the same work and fill the same offices. The only difference is, we love our slaves, and we are ready to defend, assist and protect them; you hate and fear your white servants, and never fail, as a moral duty, to screw down their wages to the lowest, and to starve their families, if possible, as evidence of your thrift, economy and management - the only English and Yankee virtues."
"Whilst we hold that all government is a matter of force, we yet think the governing class should be numerous enough to understand, and so situated as to represent fairly, all interests. The Greek and Roman masters were thus situated; so were the old Barons of England, and so are the white citizens of the South. If not all masters, like Greek and Roman citizens, they all belong to the master race, have exclusive rights and privileges of citizenship, and an interest not to see this right of citizenship extended, disturbed, and rendered worthless and contemptible."
It's pretty clear that he's arguing for a white racial aristocracy as an alternative to capitalism, and not for the enslavement of poor whites. Whether what he considered "white" is necessarily what we consider "white" is debateable- I'm pretty sure he wasn't in favour of researching family trees to elevate mixed-race slaves from their condition- but views being reprehensible doesn't remove the requirement to report them accurately.
[all emphasis is mine, not original]