I am creating the Enneper Surface in Geogebra with the following equations.
Is the surface or equation correct, and does the Enneper surface look like this? I'm confused because Wikipedia has a different image of the Enneper surface. In the equation both parameters $u$ and $v$ are varying from $-5$ to $5$.
As was mentioned in the comments, the issue is that the equations you used are not quite correct. It looks like you distributed incorrectly. They should be
$$\begin{align*} x &= \frac{u}{3}-\frac{u^{3}}{9}+\frac{v^{2}u}{3}\\ y &= -\frac{v}{3} + \frac{v^{3}}{9} - \frac{vu^{2}}{3}\\ z &= \frac{u^2}{3} - \frac{v^2}{3} \end{align*} $$
You can see the difference between the two surfaces on CalcPlot3D.
To get an plot like the one on Wikipedia it is easier to use the polar parameterization:
$$\begin{align*} x &= r\cos(\phi) - \frac{1}{3}r^{3}\cos(3\phi)\\ y &= -\frac{1}{3}r\left[3\sin(\phi) + r^{2}\sin(3\phi)\right]\\ z &= r^{2}\cos(2\phi) \end{align*} $$
Plotting this with $r\in [0,2]$ and $\phi \in [-\pi,\pi]$ will produce a plot similar to the one on Wikipedia. Again, you can see this on CalcPlot3D