The entire sentence is about "Whatever you tell me". This is the subject of the sentence. The verb phrase is "shall remain". Shall is a helping verb (think will). Remain is a linking verb. Secret is a predicate nominative (type of subject complement) that renames the subject as a secret. Ours is a possessive adjective describing whose secret it is. Since there is no action verb there is no direct object. There is also not a preposition so there cannot be an object of a preposition.
A subject refers to the person, object, event, place, idea, being, etc. (whether plural or singular) that is being discussed, described or dealt with in a sentence, and a subject may be a group of words (like phrases or clauses) or just one (E.g. "I" and "Mary"). In the sentence, "Whatever you tell me" is acting as a noun and a subject: It is the main focus in the sentence and it's what is being dealt with.
