Question: Is this operation allowed?

Going from this: 
$\left ( \frac{x^{2}+6}{x^{2}-4} \right )^{2}= \left ( \frac{5x}{4-x^{2}} \right )^{2}$

To this: $\left ( \frac{\left (x^{2}+6 \right )\left ( 4-x^{2} \right )}{\left (x^{2}-4 \right )5x} \right )^{2}= 1$

Answer: 
Yes.

In response to your edit: You squared both sides of the equation to correct your post. However, this doesn't make a difference, it's still valid. Since you squared both sides of the equation (applied the same operation), you didn't effectively change anything.

The solution is therefore:

1. take the square root
2. multiply both sides by $(x^{2}-4)$ and divide both sides by $5x$.
3. square both sides ( $1^2=1$ )
If you are solving for $x$, it's OK as long as in the end you check whether the expression you divided by, $5x$, is non-zero.

In general, you would have to follow through with everything assuming all transformations are legal, and then check for weak spots like division by zero ($5x=0$ or $x^2-4=0$, or square roots or logs of negative numbers), and modify your set of solutions accordingly by excluding some solutions that would lead to a log of a negative number, or adding some solutions that could have lead to a zero on both sides of the equation.