Can I have my subscribers unsubscribe on my own website?

No, this is not possible. Your subscribers can only unsubscribe using the unsubscribe link. They will then always see our unsubscribe message, where they can confirm their unsubscribing.

It is our responsibility to make sure that each unsubscribe is processed correctly. If your website is temporarily offline, or unsubscribing is inadvertently not installed properly, an unsubscribe from your list could not be processed. 

What you can do is put your own logo with the unsubscribe message. After unsubscribing, you can also link your subscribers to your website.