Does flying an airplane upside down imply negative AoA? Why?
Level flight (no change in altitude) upside down, with an airfoil with a zero-lift AoA of less than or equal to zero degrees, requires a negative AoA to get upwards lift. 

This is assuming the AoA is measured in the body frame of the aircraft, aligned with the chord line of the airfoil.

Also this assumes there is no thrust/drag or thrust/drag do not meaningfully contribute to vertical net forces.