Both sides agree that....universal truths of the sort expressed in the Declaration of Independence (or in twentieth-century liberalism: they tend to see the two as continuous) are ultimately destructive of authentic, historically rooted human communities;
How are the universal truths in the Declaration of Independence destructive?
What the post said is that both the Paleo and Neo point of view hold the "We hold these truths to be self evident" declaration destructive. The post did not say that "these truths" were destructive. This is a meaningful difference. Perhaps you meant to ask "How can the Paleos and Neos hold the 'these truths' declaration to be destructive?" That I can answer.
The Neos say that the "these truths" statement is a declaration of faith, a declaration that persons are "endowed by their Creator", that "these truths" is a declaration of belief in God, that is, that human rights rest upon belief in God, in faith in Him, and in essence nothing else. Neos are mostly Athiests and so see this as no basis what so ever for human rights and therefore look for some other basis for human rights.
The Paleos mostly hold that the "these truths" declaration is a Deist corruption of revealed religion. Jefferson was by no means a believing Christian (nor Franklin and others) and many Paleos hold against Jefferson that he was apparently comfortable putting words into the mouth of God. Most Paleos believe nothing good can come of this practice, which in Christianity is a very serious sin indeed.