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We can split this question into 3 cases:

Case 1: $\dfrac{n^2-1}{5}-1\equiv0\space\text{mod 9}$

Case 2: $\dfrac{n^4+3n^2-4}{25}+1\equiv0\space\text{mod 9}$

Case 3: $\dfrac{n^2-1}{5}-1\equiv\dfrac{n^4+3n^2-4}{25}+1\equiv0\space\text{mod 3}$

Now, we shall consider each of these cases. For case 1:

$\dfrac{n^2-1}{5}-1\equiv0\space\text{mod 9}$ $\dfrac{n^2-1}{5}\equiv1\space\text{mod 9}$ $n^2-1\equiv5\space\text{mod 9}$ $n^2\equiv6\space\text{mod 9}$ This implies that $n$ is divisible by 3 but $n^2$ is not divisible by 9, a contradiction. Hence, there are no solutions for this case.

Now, we shall consider case 2. $\dfrac{n^4+3n^2-4}{25}+1\equiv0\space\text{mod 9}$ $n^4+3n^2\equiv6\space\text{mod 9}$ $n^2(n^2+3)\equiv6\space\text{mod 9}$ This implies that the product of $n^2$ and $n^2+3$ is divisible by 3, so at least one of them is divisible by 3. However, since both have the same remainder modulo 9, so both of them are divisible by 3. This means that product is divisible by 9, a contradiction. Hence, there are no solutions for this case.

Finally, we shall consider case 3. $\dfrac{n^2-1}{5}-1\equiv0\space\text{mod 3}$ $n^2\equiv0\space\text{mod 3}\Rightarrow n^2\equiv0\space\text{mod 9}$ Also, note that $n^2\equiv1\space\text{mod 5}$ in order for it to be an integer. Furthermore, we have $\dfrac{n^4+3n^2-4}{25}+1\equiv0\space\text{mod 3}$ $n^4+3n^2-4\equiv2\space\text{mod 3}$ $\therefore n\equiv0\space\text{mod 3}$ Also, note that $n^4+3n^2-4\equiv0\space\text{mod 25}$ in order for it to be an integer. $(n^2-1)(n^2+4)\equiv\text{0 mod 25}$ Note that the two terms above have the same remainder modulo 5, so solving, we also get $n^2\equiv\text{1 mod 5}$ .

Hence, after considering all three cases, we get the following conditions for $n$ : $n^2\equiv1\space\text{mod 5}$ $n^2\equiv0\space\text{mod 9}$ Solving, we get $n^2=9(5t+4)$ for some integer $t$ . Note that in order to satisfy the perfect square condition, $5t+4$ is a perfect square. Hence, trying out numerous values of $t$ , we get the desired values, $\huge\text{6, 9, 21, 24, 36.}$