Consider the standard embedding of the unit interval in $\mathbb R^2$ viz. $I=[0,1]\times \{0\}
\subset \mathbb R^2$. Let $C$ denote the Cantor subset $C \subset I$ and define $U= \mathbb R^2 - C$, an open subset of $\mathbb R^2$. 

I seem to remember that $\pi_1(U)$ has cardinality at least the continuum and so the fibers of the universal covering $\tilde{U} \to U$ are such  big discrete sets that I would guess that $\tilde{U} $ can't be embedded in $\mathbb R^2$.