you need to add c1,c2,c3,c4 to the variables

In[5]:= Reduce[{c1 == x1 x3, c2 == x1 x4, c3 == x2 x3, 
  c4 == x2 x4}, {c1, c2, c3, c4, x1, x2, x3, x4}]

Out[5]= (c1 == 0 && c2 == 0 && c3 == 0 && c4 == 0 && x1 == 0 && 
   x2 == 0) || (c1 != 0 && c4 == (c2 c3)/c1 && x2 == (c3 x1)/c1 && 
   x1 != 0 && x3 == c1/x1 && x4 == (c2 x3)/c1) || (c1 == 0 && 
   c2 == 0 && c3 == 0 && c4 == 0 && x1 != 0 && x3 == 0 && 
   x4 == 0) || (c1 == 0 && c2 == 0 && x1 == 0 && x2 != 0 && 
   x3 == c3/x2 && c3 != 0 && x4 == (c4 x3)/c3) || (c1 == 0 && 
   c3 == 0 && c2 != 0 && x2 == (c4 x1)/c2 && x3 == 0 && x1 != 0 && 
   x4 == c2/x1) || (c1 == 0 && c2 == 0 && c3 == 0 && c4 == 0 && 
   x1 == 0 && x2 != 0 && x3 == 0 && x4 == 0) || (c1 == 0 && c2 == 0 &&
    c3 == 0 && x1 == 0 && x3 == 0 && x2 != 0 && x4 == c4/x2 && 
   c4 != 0)