If I move out of a battlefield I conquered and played a hidden card there, does the hidden card disappear?

Yes — if you move your last unit out of that battlefield, the hidden card goes to your trash.

A Move triggers a Cleanup when it completes (Rule 453). If you have no units left at a battlefield in an Open State, you lose control of it in that cleanup (Rule 323.6), and any hidden cards at battlefields you no longer control are placed in their owner's trash (Rule 323.7, 107.3.d; FAQ #6608). Control is only preserved if another of your units remains there, or if a Showdown/Combat is ongoing at that battlefield.

So the card only survives the move if you keep control — leaving even one unit behind keeps the hidden card safe.