is a negative Kc possible. Also, is a negative Ka possible?
No and no. But let's reason this out. First time I've been asked this question, ever.
For the reaction A + B ==> C + D,
Kc = [A][B]/[C][D]
If A and B are as small as they can be (which would be zero), and C and D are as large as they can be (which would be a huge number like infinity), then Kc would be zero. Since A and B can't be less than zero, Kc can't be less than zero.
For an acid, like acetic acid,
CH3COOH ==> H^+ + CH3COO^-
Ka = (H^+)(CH3COO^-)/(CH3COOH).
If H^+ is zero and CH3COO^- is zero (that is if zero ionization occurs), then CH3COOH is 1 (it is defined as 1 when we have the pure substance), 0/1 = 0 so that's as small as Ka can be. Therefore, it can't be negative either.