What happens to the units when squaring a variable?

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.

A slight expansion of the question:

When a physical quantity in an equation is raised to a power (like being squared) then all the physical quantities that go into that quantity are also raised to the same power.

So, velocity, $v$, has dimensions of length over time, $l/t$, and velocity squared, $v^2$, has dimensions of length squared over time squared, $l^2/t^2$.

As a consequence of this, the units used for these quantities follow the same rules.

Additionally, in any physics formula the physical quantities on the right hand side must equal the physical quantities on the left hand side.

Google "dimensional analysis"

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:

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

I think your all wrong. You can not square a velocity. It is nonsense. Draw me a illustration of a squared constant velocity. It's not possible. If it is a constant velocity and over any unit of time you can not produce a square.. Especially the speed of light. The speed of light is the product of a circle . Light radiates in all directions . If it radiates out say for 1 second it will create a circle and you can not make it make a square in one we one or na million light years . You can not square a circle. Take two 4 inch lines make one a square and one a circle . Calculate the area of the circle and the square. Now tell me that the square of velocity is not a circle. The square of the velocity of light is wrong it's not possible.. what ever.