There is no such $c$ even if we use only $2 \times 2$ matrices.
For any $c \geq 1$ let $A,B$ be the positive-semidefinite matrices
$$
A = \left( \begin{array}{lc} c^2 & c \cr c & 1 \end{array} \right),
\phantom\infty
B = \left( \begin{array}{cc} 1 & 0 \cr 0 & 0 \end{array} \right).
$$
of rank $1$. Then we calculate that the difference
$$
D := c(A+B)^2 - (A-B)^2
$$
has determinant $(c-1)^2 - 4c^3 < 0$, and is thus not positive semidefinite.

In fact this counterexample works for all $c \in \bf R$:
looking around $\ker A = {\rm span} \lbrace(-1,c)\rbrace$
we find the negative vector $v = (-c, c^2+1)$.
To verify that $\langle v, Dv \rangle < 0$, recall that
for any vector $x$ and any symmetric matrix $M$ of the same order we have
$$
\langle x, M^2 x \rangle = \langle Mx, Mx \rangle = |Mx|^2.
$$
Here we compute $v(A+B) = (0,1)$ and $v(A-B) = (2c,1)$, so
$$
\langle v, Dv \rangle = c \phantom. |(0,1)|^2 - |(2c,1)|^2 = -4c^2 + c - 1,
$$
which is negative for all real $c$. If $c$ is small enough that
$\det(D) > 0$ then $D$ is *negative* definite.