What happens to the units when squaring a variable?

Yes. If you square a variable, its unit of measurement is also squared, in the case of speed $v$ in $m/s$ ($ms^{-1}$), then $v^2$ is expressed in $m^2s^{-2}$. This is true for all physical variables (or constants).

Yes. Consider the equation for kinetic energy (KE):

$${\rm KE} = \frac{1}{2} mv^{2}$$

the dimensions of KE are:

$${\rm mass} \times {\rm velocity}^{2}=\frac{{\rm mass} \times {\rm length}^{2}}{{\rm time}^{2}}$$

or with SI units:

$$1\,{\rm J} = 1\,{\rm kg}\,{\rm m}^{2}\,{\rm s}^{-2}$$

Yes.The unit of $(\text{velocity})^2$ is $[\frac{\text{m}}{\text{s}}]^2$ .This is true for all calculations for any physical quantity.On squaring a physical quantity, its dimension gets squared. As a result, the unit is also squared.