How do I check for linearity for the following piecewise-defined system?
First, you are not trying to find the linearity of a signal, but the linearity of a system defined by the transform above between input and output signals.
If you suspect your system is not linear (the different expressions depending on the sign of $x(t)$ is a big red flag), then you only need a counterexample rather than a general proof of linearity.
In this case, if you scale your input by a negative number, the output is not scaled accordingly. That is: $T{\alpha x} \neq \alpha T{x}$ if $\alpha$ is negative.