The item that desperately asks you not sell it is more than likely a dummy of some kind, used for assigning fighters or hiring heroes as the New Republic.

Selling it is very bad, I would suggest you reload a old save or restart it.

I attacked Don't sell me with the Viscount Prototype and 4 Bothan Cruisers, managed to defeat "Assign Wedge Antilles's Rogue Squadron", "Assign Squadron", and "View Current Assignment", but I lost all my space stations.
Worth it.

EDIT: Should have heeded Hawkeye's advice.
Having defeated the "Assign" icons, I can no longer change pilot assignments.