Acne is an immunological disease. It's not just oxidation of lipids. A couple minor points: the rabbit ear is a woefully inadequate model for human acne due to the prevalence of false positive results (and differing circumstances, like lack of bacterial colonization).
Also, squalene and other lipids are oxidized in everyone's skin whenever they go into the sun, and if anything, sun has been shown to help acne, probably because oxidized squalene increases inflammation which results in faster lesion healing (not necessarily reduced lesion occurrence). Older people also have plenty of oxidized squalene, probably more than younger people, and yet they have less acne. No causation has been shown here.
On the other hand, there is certainly truth behind the idea that acne is an inflammatory disorder, which is why pretty much every single acne medicine has an anti-inflammatory effect. And antioxidants can be anti-inflammatory either directly or indirectly, thus exerting similar effects. But it's important to realize here that these antioxidants would not be addressing the cause of the problem, as most acne medications fail to do. The cause of the problem is a fundamentally dysregulated immune system; the body reacts to a commensal organism or food antigen in a way that it shouldn't. And this is very hard to fix.