The question asks for a bounded linear operator on a Hilbert space satisfying the condition in the title. This is what I came up with:
Let $A_1:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a 90-degree rotation (counterclockwise, say, it doesn't matter) and let $A_2:\mathbb{R}^2\rightarrow \mathbb{R}^2$ map $e_1$ to $(1/2)e_1$ where $B = \{e_1, e_2\}$ is the standard basis for $\mathbb{R}^2$. Then let $A = A_1 A_2$. Then $\|A\| = .5$.
Am I correct that $\|A^2\| = .5$ as well?
(I've tagged this question "examples-counterexamples" because it seems to me to be an example of equality in the Cauchy-Schwarz inequality for the operator norm.)
Your description of your example is somewhat verbose compared to just writing down the matrix: $A=\begin{bmatrix}0&-1\\1/2&0\end{bmatrix}$. It doesn't have $\|A\|=0.5$ as you claim, though, because $Ae_2=-e_1$. In fact $\|A\|=1$, but since you do have $A^2=-\frac12I$ with norm $\frac12\ne 1^2$ it does work as an example.
A simpler example would be $A=\begin{bmatrix}0&0\\1&0\end{bmatrix}$, where again $\|A\|=1$ but $A^2=0$.