What, if anything, is a square radian? My 1st year Mathematics BSc course notes on circular motion use
\begin{align}
\frac{d}{dt}(\sin\theta)
&amp;=\frac{d}{dt}(\sin(\omega t))\tag{1.1}\\
&amp;=\omega\cos(\omega t),\tag{1.2}\\
\frac{d^2}{dt^2}(\sin\theta)
&amp;=\frac{d^2}{dt^2}(\sin(\omega t))\tag{2.1}\\
&amp;=\frac{d}{dt}(\omega\cos(\omega t))\tag{2.2}\\
&amp;=-\omega^2\cos(\omega t),\tag{2.3}
\end{align}
where


$\omega$ is angular speed, a scalar, with units $\text{rad}\cdot s^{-1}$;
$t$ is time, a scalar, with units $s$.


From this it seems to me to follow that $\omega^2$ has units $(\text{rad}\cdot s^{-1})^2=\text{rad}^2\cdot s^{-2}$. Is that right, and, if so, does a square radian have a physical meaning?
"rad(ians)" is just another word for the number $1$, a unit-less unit. An angle is some part of a full rotation at $2\pi$, a unitless number. However, to tell that any number stands for an angle and not some amount of apples or cake, there is the habit to add "radians" to it.
OP answering own question.

Remarks by Hagen von Eitzen and Lutz Lehmann have prompted me to think of the following answer.

A radian is a dimensionless fraction with value 
$$\frac{1}{2\pi}.$$

It seems to me to follow that a square radian is a dimensionless fraction with value 
$$\left(\frac{1}{2\pi}\right)^2=\frac{1}{4\pi^2}.$$

This can be thought of as the angle by which the angular speed increases each second.