How to find $\lim_{n \to \infty}\int_{0}^{1}\sin^2\left(\frac{1}{ny^2}\right)\,\mathrm{d}y$ if it exists?
After a change of variables the integral becomes $$I_{n}=\frac{1}{2}\int_{1}^{+\infty}\sin^2 \bigg(\frac{x}{n}\bigg) x^{-\frac{3}{2}}dx$$ Substituting $s=\frac{x}{n}$ we get $$I_{n}=\frac{1}{2\sqrt{n}}\int_{\frac{1}{n}}^{+\infty}\frac{\sin^2(s)}{s^{\frac{3}{2}}}ds \leq \frac{1}{2\sqrt{n}}\int_{0}^{+\infty}\frac{\sin^2(s)}{s^{\frac{3}{2}}}ds$$ And since $I_{n} \geq 0$ and the last integral is just a constant we conclude by the squeeze theorem $$\lim_{n \to +\infty} I_{n}=0$$
Probably too complex
For the antiderivative first $$I_n=\int\sin^2\left(\dfrac{1}{ny^2}\right)\,dy$$ One integration by parts gives $$I_n=y \sin^2\left(\dfrac{1}{ny^2}\right)+\int\frac{2 }{n y^2}\sin \left(\frac{2}{n y^2}\right)\,dy$$ The remaining integral is computable in terms fo Fresnel sine integral.All of that makes $$I_n=y \sin ^2\left(\frac{1}{n y^2}\right)-\frac{\sqrt{\pi } S\left(\frac{2}{ \sqrt{n\pi } y}\right)}{\sqrt{n}}$$
Using the bounds $$J_n=\int_0^1\sin^2\left(\dfrac{1}{ny^2}\right)\,dy=\frac{-2 \sqrt{\pi } S\left(\frac{2}{ \sqrt{n\pi }}\right)+\sqrt{n}-\sqrt{n} \cos \left(\frac{2}{n}\right)+\sqrt{\pi }}{2 \sqrt{n}}$$
Expanding for large values of $n$ $$J_n=\frac{1}{2} \sqrt{\frac{\pi}{n}}-\frac{1}{3 n^2}+O\left(\frac{1}{n^4}\right)$$
Checking for $n=10$ using numerical integration $I_{10}=0.276921$ while the above truncated expansion gives $0.276916$.
Let the integral under limit be denoted by $I_n$. Since the integrand is non-negative we have $I_n\geq 0$.
Let's take an $\epsilon$ with $0<\epsilon<1$ and split the interval of integration into $[0, \epsilon] $ and $[\epsilon, 1]$ so that the integral $I_n$ is split as sum of two integrals. Since the integrand is bounded above by $1$ the first integral does not exceed $\epsilon $. Since $\sin^2x\leq x^2$ the second integral does not exceed $$\int_{\epsilon} ^{1}\frac{dy}{n^2y^4}=\frac{1}{3n^2}\left(\frac{1}{\epsilon^3}-1\right)$$ Therefore we have $$0\leq I_n\leq \epsilon+\frac{1}{3n^2}\left(\frac{1}{\epsilon^3}-1\right)\tag{1}$$ for all $n$ and all $\epsilon\in(0,1)$. Letting $n\to\infty$ we can see that $$0\leq \liminf_{n\to\infty} I_n\leq \limsup_{n\to\infty} I_n\leq \epsilon$$ Since $\epsilon\in(0,1)$ is arbitrary it follows that the desired limit is $0$.
As mentioned in comments, you can put $\epsilon =1/\sqrt{n}$ in the inequality $(1)$ and apply usual Squeeze theorem to get the desired result.