Integer solution $(x,y,z)$ to $xyz=24$

Your logic is correct for positive solutions. Now you need the sign pattern to be $+++,+--,-+-,--+$, four possibilities, so multiply by four.

Yes, this looks right for finding the number of positive integer solutions.

Since you originally asked for all integer solutions, you'll also have to take into account the signs of the factors. There are $\binom{3}{0} + \binom{3}{2} = 4$ ways of adding zero or two minus signs, for a total of 120 integer solutions.