The Wikipedia article on Antifeminism is very limited.
In my opinion the natural antonym of feminism isn't "anti-feminism": it's sexism -- which I understand to mean, approximately, that people's role and status in society and social relationships is determined by their gender.
Many religions preach that men and women are fundamentally different, and even that men are superior to women. Such religions oppose feminism.
I'm not sure I agree.
Instead I think there are many people who preach that, but ... when slavery (for example) was a widespread practice, people preached (I'd say "invented") religious justifications for that. I see the "religious" justification as poor excuses though, rather than the reasons (i.e. "why?") that you may be asking about.
I am looking specifically for non-religious reasons cited by non-feminists describing why they oppose feminism [the aims of feminism and not just the means of achieving equal rights].
I'm not sure you'll find them.
Consider this analogy: imagine there were a movement towards financial equality -- e.g. so that poor people become richer.
- Will anyone speak out against that aim? No, they won't -- it's the American dream, this is the "land of opportunity", and so on.
Will anyone speak out against the methods? Yes for sure, they do -- "I'm alright, Jack" -- people oppose taxes (including income, wealth, and estate taxes), public education, the "welfare" state, public health insurance and/or health care, reverse discrimination or affirmative actions, labour regulations (minimum wage, parental leave), the universal wage is a pipe-dream, and so on.
Incredibly, some people oppose it on religious grounds -- e.g. prosperity theology (see also "protestant work ethic" and Achievement ideology) ... and/or, invoking the spectre of godless communists (it's hard for me to imagine how socialism came to be viewed as a- or anti-religious) -- but I fail to see religion as a motive, just as a poor excuse (I think it's just an exercise of "might makes right" ... or as La Fontaine put it in the 17th century, la raison du plus fort est toujours la meilleur, "the reasoning of the stronger is always superior", also known as ultima ratio regum).
Speaking of "kings", see e.g. the web site Return of Kings (I won't link to it). I take that as an example of something called, more generally the Men's rights movement (Wikipedia quotes sources describing it as "antifeminist" and as advocating "male supremacy" and misogyny) -- I don't think that's religious at all, except perhaps where it allies with anti-abortion advocates (incidentally I'm not sure that "anti-abortion" is a religious principle either, see e.g. Christians in the Hand of an Angry God (part 4), which starts, "So what does the Bible say about the intentional termination of a pregnancy? Nothing. That's right, nothing. It never comes up. Even in the holiness code, which takes time out to preach about the evils of mildew, there isn't a single thing about the intentional termination of a pregnancy.", and goes on to argue that life or at least the soul begins at birth with the first breath and that a fetus might be considered like property ... that whole screed, in 5 parts, is more generally about how "religion" came to be linked with "right-wing" politics in America).
Also apparently there are some men who believe that women are privileged, in society, and that men are relatively exploited and socially conditioned to be subservient. To the extent that feminism may propose (even) more advantages to women, I suppose these men would be against that aim. I can only offer this link as a reference to that -- What is the red pill philosophy? -- because I'm unfamiliar with their writings.
Another kind of antifeminism (apart from outright male supremacy or dominance), which also isn't necessarily religious (and which may be even more widespread than male supremacy, e.g. I imagine that successful and/or traditionally-educated women, too, find it appealing) comes from Social conservatism -- who, I suppose, argue that traditional gender roles are good. I think this isn't necessarily antifeminist -- "Marry, be a wife and mother, stay at home ... or don't, it's up to you!" is a type of equality -- but (e.g. per that Wikipedia article) I think it often is (and may also be opposed to the "aim" of feminism and not just the methods).
There's some link between conservatism and religion too (e.g. religion existed in the past, is traditional).
Finally I suppose there are people whose view is that "equal rights" exist already (and that therefore, I don't know, feminists advocating change are intent on disrupting the lives of others, making a name for themselves for some useless/trivial/invented reason: see also "grievance politics","victim mentality", "identity politics"). On that subject I note, too, that the OP summarises feminism as "equal rights for men and women" -- which ("rights") might restrict the topic to "theoretical legal rights" rather than "actual social opportunities" (allowing you to oppose anything more than "equal rights" as political "overreach") ... and which ("men and women") might be heteronormative, and exclude equal opportunities for children especially including girls.
P.S. It isn't only a matter of politics and economics (and sociology), it's also a matter of interpersonal relationships -- e.g. I presume that feminists oppose criminal sexual harassment, however the whole topic of "harassment" is controversial and criticised ("Though the phrase sexual harassment is generally acknowledged to include clearly damaging and morally deplorable behavior, its boundaries can be broad and controversial. Accordingly, misunderstandings can occur"). Therefore I suppose that feminism (or feminists) might be viewed as wanting to police sexual behaviour -- "innocent flirting" too, for example -- or conversely, to license it. Wikipedia says "In most modern legal contexts, sexual harassment is illegal" but it also says that wold-whistling might be considered "a form of sexual harassment in itself" -- I say this to show that the boundaries maybe aren't clear. I suppose that there's "illegal harassment" but also "legal (or not-illegal) harassment", that feminists might be anti both forms, and that that is not entirely popular.