The open subset $U$ is parallelizable and hence so is its universal cover. A classical theorem of Morris Hirsch says that any open parallelizable $n$-manifold can be immersed into $\mathbb R^n$. Now one could ask whether any open parallelizable $n$-manifolds embeds into $\mathbb R^n$. This is formally more general than the original question, so it might be easier to produce a counterexample in this case. Also this more general question strikes me as more natural.

The open subset $U$ is parallelizable and hence so is its universal cover. A classical theorem of Morris Hirsch says that any open parallelizable $n$-manifold can be immersed into $\mathbb R^n$. Now one could ask whether any open parallelizable $n$-manifold embeds into $\mathbb R^n$. This is formally more general than the original question, so it might be easier to produce a counterexample in this case. Also this more general question strikes me as more natural.

The open subset $U$ is parallelizable and hence so is its universal cover. Now a classical theorem of Morris Hirsch says that any open parallelizable $n$-manifold can be immersed into $\mathbb R^n$. Whether it can be embedded seems to be a subtle matter; in fact I do not see any other general topological restrictions on open subsets of $\mathbb R^n$.

The open subset $U$ is parallelizable and hence so is its universal cover. A classical theorem of Morris Hirsch says that any open parallelizable $n$-manifold can be immersed into $\mathbb R^n$. Now one could ask whether any open parallelizable $n$-manifolds embeds into $\mathbb R^n$. This is formally more general than the original question, so it might be easier to produce a counterexample in this case. Also this more general question strikes me as more natural.