For fixed $n$ and $k<n$ the distance $|X_k-Y_k|$ is uniformly distributed in $[0,1/(2n)]$. Thus if $\alpha_n<1/(2n)$, then $$P\left[\max_{1 \le k \le n}|X_k-Y_k| \le \alpha_n \right]=(1-2n\alpha_n)^n \,. $$ On the other hand, if $\alpha_n \ge 1/(2n)$, then this probability is clearly 1. So this gives a negative answer to the question.

For fixed $n$ and $k \le n$ the distance $|X_k-Y_k|$ is uniformly distributed in $[0,1/(2n)]$. Thus if $\alpha_n<1/(2n)$, then $$P\left[\max_{1 \le k \le n}|X_k-Y_k| \le \alpha_n \right]=(1-2n\alpha_n)^n \,. $$ On the other hand, if $\alpha_n \ge 1/(2n)$, then this probability is clearly 1. So this gives a negative answer to the question.