I'm pretty sure Postbox would tick all of those boxes. I definitely use non-GMail accounts and multiple signatures - each account has a different default signature, and I can change the signature on individual messages on the fly.
Not sure about encryption, but it's bound to have a PGP plugin or something similar. Postbox is partly derived from Thunderbird, so it would probably satisfy your requirements as well.
Edit: Just realized that I've assumed Apple Mail refers to the Mac version. I'm not sure about iOS, but the new Microsoft Outlook (rebranded Acompli) might do some of those things. Personally I use mobile only for triage, and write emails when I'm on my laptop.