The question might seem profoundly ambiguous (to anyone who isn't an immigration lawyer, which I am not) but I think the keywords "resident alien" and "permanent employment" in the same sentence suggest what they are looking for.
Note that US tax law and immigration law are quite independent; it is possible to be what tax law might call a "resident alien" (or a "US person") while having no legal immigration status in the country at all. For the purposes of legally accepting employment, however, tax law is irrelevant and it is the definition of "resident alien" in immigration law that matters. The definition that USCIS uses is here, where it basically refers to someone with a green card.
I believe the odd wording about "eligible" and "permanent employment" is meant to qualify not only green card holders but those with Employment Authorization Documents who are free to take any work they can find. People with an EAD are generally (but not always) the beneficiaries of an application for immigrant status which is pending but not yet approved. My guess would be that the "permanent employment" part is meant to disqualify those who would need to acquire a nonimmigrant visa to do the work since nonimmigrants can only accept work for a limited term (TN applicants are sometimes disappointed when the offer letter is for a "permanent" position).
I hence think the correct answer to that question in your case would be "No", since that would tell the employer that they need to sponsor you for a nonimmigrant status and I think that is what the asker is trying to find out.