Neither of those is great, with the first being especially confusing.
There is a confusion here, but it is not about the grammatical number so much as it is about whether your negation applies to the items separately or jointly.
Imagine at a summer barbecue someone offers you a sandwich hot off the grill, where hot dogs and ham patties are sizzling away. Consider then these choices:
- I don’t want a hot dog or a ham patty because I don’t eat it.
- I don’t want a hot dog or a ham patty because I don’t eat them.
- I don’t want a hot dog or a ham patty because I don’t eat either (of those).
- I don’t want a hot dog or a ham patty because I eat neither (of those).
The first is confusing because we don’t know which one you’re saying you don’t eat.
The second is better but still potentially confusing because we wonder whether you’re saying you don’t eat both of them at the same time, whether in one sandwich or in succession.
No such confusion exists in the third or fourth choices, because by using either or neither you make clear that you are negating both conditions. You don’t eat hot dogs and you also don’t eat ham patties: not one at a time, not two at a time, you do not like them, Sam-I-am. :)