Under steady state conditions, it is not expected that the heat transfer coefficient would be negative. Unless there is an unsteady state process involved. Another reason could be if other processes influencing the heat flux are not considered such as internal reactions, frictional heating, JT effect, radiation or mass transfer for evaporation.

I don't think you should expect a negative heat transfer coefficient under any condition.

If you are deducing the heat transfer coefficient from experiments, and you define Newtons law as q"=h*(Tfluid-Tsurface), then h will become negative as Tfluid < Tsurface. Otherwise, h should be positive at steady-state unless there are some additional heat source/sinks not accounted for.