Pluses and minuses all the way!

The expansion of $\displaystyle \sum_{k=0}^{28} (-1)^k a_k = a_0 - a_1 + a_2 + ...+a_{28}$ has 15 $+$ signs (when $k$ is even) and 14 $-$ signs (when $k$ is odd). Then $\displaystyle \left(\sum_{k=0}^{28} (-1)^k a_k\right)^2 = \sum_{k=0}^{28} a_k^2 + 2 \color{#3D99F6}{\sum_{i \ne j}^{28} (-1)^{i+j}a_ia_j}$ . The $-$ signs only come from the second summation (blue) and when $i+j$ is odd or either $i$ or $j$ is odd and the other is even. Since there are 15 even $k$ 's and 14 odd $k$ 's, there are $15 \times 14 = \boxed{210}-$ signs.

Each term in the expansion consists of two summands. There are 14 negative signs and 15 positive signs. A term has a negative sign in the expansion if it is a product of unlike signs. Hence, there are $(14)(15)=210$ negative signs.