What is the ruling on saying "amanah 'alayk" (I beg you by your honor/trust) or "Allah 'alayk" (I beg you by God) to refrain from a certain action or revealing a secret, and are these phrases binding?

It is necessary to avoid saying "I entrust you with it" (أمانة عليك) because it is possibly an oath by the trust, which is forbidden. This is unless the intention behind it is that this matter is a secret and a trust that must be concealed. As for saying "by God on you" (الله عليك), it appears to be an oath by God, and there is no harm in it. It is recommended to fulfill the request of one who swears by God, unless doing so would lead to corruption.