Yes, you can leave OWA as HTTP-only, however this presents a significant security risk as you're passing a user token that could potentially be intercepted and used to impersonate the user. You also may get mixed-content warnings in IE if mixing SSL (SharePoint)
and HTTP (OWA).
Apps should be done over SSL because, again, you're sending NTLM hashes which is extremely insecure.
Trevor Seward
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This post is my own opinion and does not necessarily reflect the opinion or view of Microsoft, its employees, or other MVPs.