show that $ \limsup n\; | \;\{ (n+1)^2 \sqrt{2}\} - \{ n^2 \sqrt{2}\}\; | = \infty $

One have $(n+1)^2=n^2+2n+1$ and so $\{(n+1)^2\sqrt 2\}= \{\{n^2\sqrt2\}+\{(2n+1)\sqrt2\}\}$. $\{(n+1)^2\sqrt 2\}$ is close to $\{n^2\sqrt 2\}$ only if $\{(2n+1)\sqrt 2\}$ is close to $0$ or $1$. However $\{(2n+1)\sqrt 2\}$ lies in $[\frac{4}{10};\frac{6}{10}]$ infinitely often so $|\{(n+1)^2\sqrt2\}-\{n^2\sqrt2\}|\geq \frac{4}{10}$ infinitely often.

Note that this is not an if and only if situation, for example if you replace $n^2$ by $\sqrt n$ the $\limsup$ is still infinite even if $\{\sqrt{n+1}-\sqrt n\}$ will approach $0$ when $n\to \infty$. This is of course because $|\{\sqrt{n+1}\}-\{\sqrt n\}|$ is big when $\sqrt n < k < \sqrt{n+1}$ for some integer $k$, and that happens infinitely many times.

Note also that this doesn't use the equirepartition of $(n^2\sqrt2)_{n\in \mathbf N}$, it merely use the equirepartition of $((2n+1)\sqrt2)_{n\in \mathbf N}$.

Let $x_n$ be a sequence and call $d_1(n)$ the number of times $\{x_k\}$ is in $I_1 = [0;\frac 14]$ when $k \in \{1 \ldots n\}$, and $d_2(n)$ the number of times they are in $I_2 = [\frac 12 ; \frac 34]$.

Let $\epsilon \in (0 ; \frac 18)$.

If $(x_n)$ is uniformly distributed mod $1$ then $d_1(n)/n$ and $d_2(n)/n$ converge to $\frac 14$ as $n$ gets larger, so there is an integer $m$ such that $|d_i(n)/n - \frac 14| < \epsilon$ for $n \ge m$.

Then, $d_i(m) < (\frac 14+\epsilon)m < d_i(\frac{1+4\epsilon}{1-4\epsilon}m)$.

and so there must be for both $i$ at least one index $n_i$ between $m$ and $m' =\frac{1+4\epsilon}{1-4\epsilon}m$ such that $\{x_{n_i}\} \in I_i$.

Then, letting $A = \max_{m < n \le m'} \{n|x_n-x_{n-1}|\}$, and supposing $n_1 < n_2$ for convenience,
$\frac 14 \le |x_{n_1} - x_{n_2}| \le A(\frac 1{n_1+1} + \ldots \frac 1{n_2}) \le A \log(\frac{n_2}{n_1}) \le A \log(\frac {m'}m) = A \log (\frac {1+4\epsilon}{1-4\epsilon}) \le 8A\epsilon\log 3$
(the last inequality comes from $\epsilon < \frac 18$ and a bit of analysis )

This proves that $A \ge C/\epsilon$, where $C = 1/(32\log 3) > 0$.

By taking smaller and smaller values for $\epsilon$ this shows that $(n|x_n-x_{n-1}|)$ must have larger and larger values, and so it must be unbounded.