How to prove that $[2^n \sqrt{2}],[2^{n+1} \sqrt{2}],\ldots, [2^{2n} \sqrt{2}]$ contains at least one even numbers for every integer $n\ge 1$?

As you observed, $2^{n-1}\sqrt2$ would have to be pretty close to an integer $y$ (in your notation), so that $|2^{n-1}\sqrt2-y|<\frac1{2^{n+1}}$ and with $q:=2^{n-1}$, $$\tag1\left|\sqrt 2-\frac{y}{q}\right|<\frac14\cdot\frac1{q^2}. $$ In particular, $$\tag21<\frac yq<2.$$ We know that $\sqrt2$ is a root of the irreducible polynomial $f(X)=X^2-2$, hence by the IVT, $$\tag3f(\tfrac yq)=f(\tfrac yq)-f(\sqrt 2)=(\tfrac yq-\sqrt2)f'(\xi)$$ for some $\xi$ between $\sqrt2$ and $\frac yq$; from (2), we see that $1<\xi<2$. As $f(\tfrac yq)$ is a nonzero(!) integer multiple of $\frac 1{q^2}$ and as $|f'(\xi)|=|2\xi|<4$, we have $$\left|\sqrt 2-\frac yq\right|=\frac{|f(\tfrac yq)|}{|f'(\xi)|}>\frac{\frac1{q^2}}4 $$ contradicting $(1)$.

By a basic trick of Diophantine approximation we have, for all positive integers $p,q$, the inequality $$ \left|\frac pq-\sqrt2\right|\cdot\left|\frac pq+\sqrt2\right|=\left|\frac{p^2}{q^2}-2\right|\ge\frac1{q^2}. $$ In particular, if $3/2>p/q>\sqrt2$, we get the estimate $$ \left|\frac pq-\sqrt2\right|\ge\frac1{3q^2}. $$ Therefore, with $M=[2^{n-1}\sqrt2]$ we get, using $x=\{2^{n-1}\sqrt2\}$, $$ 1-x=2^{n-1}(\frac{M+1}{2^{n-1}}-\sqrt2)\ge\frac1{3\cdot2^{n-1}}>\frac1{2^{n+1}}. $$ Therefore $x<1-2^{-(n+1)}$, so your contrapositive case cannot happen.