Can a reaction have negative activation …

The difference between the maxima and the minima is the activation energy, depending on which way you’re going, the activation energy will be different.
There are instances where the activation energy can be negative. There are reactions where the rate of the reaction decreases with increasing temperatures. This happens because the when you increase the temperature, it can sometimes reduce the probability of molecules colliding as the increased momentum carries the molecules away from the potential “collision zone.” So when you fit the rate constant into an Arrhenius expression, it results in a negative activation energy. But these reactions are generally the ones without barriers. So there’s no activation energy per se.

Activation energy is always a positive quantity.It never have a negative energy.

Yes sometime it is negative in case of backward reaction 

Reaction never have negative activation energy.Its always a positive quantity.