Does the person paying the membership subscription have to be eligible for Gift Aid?

Yes. The rule for Gift Aid on membership subscriptions is the same as the rules for donations. The individuals must not pay less Income Tax and/or Capital Gains Tax than the amount of Gift Aid claimed on all their donations and membership subscriptions (not just the ones made to your charity) in that tax year. Otherwise, they will be liable to, and responsible for, repaying any difference to HMRC.