No, specific heat of a gas can't be negative. It is always positive.

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   Perhaps not..  for gases, their temperature increases as they absorb heat  and they lose heat when temperature decreases.  But for solids that are bound by gravitational energy, negative specific energy is possible.
   In general in our real world of physical quantities in homogeneous systems that follow thermodynamic equilibrium principles,  the specific heat is positive.  So systems absorb heat and their temperature increases.

   In certain non-homogeneous systems, like some radiating stars and black holes we find that specific heat can be negative.  It seems black holes become colder when they absorb mass or energy.  Their high gravitational potential energy (negative energy) makes it that way.  When radiating stars (burning stars) lose energy and mass, they become hotter and hotter and finally burn up totally.  The temperature continuously increases.  The main reason is the large negative gravitational potential energy of the star. 

   For a gas the gravitational potential energy is much less compared to kinetic energy due to high speed movement of the molecules.  So specific heat is positive.

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