Does the graph with only one vertex have an Eulerian path? And, does it have a Hamiltonian path?
I'm not sure if all graph theory books treat degenerate cases the same way, but Diestel's Graph Theory, at least, allows a path to have length $0$, i.e., to consist of a single vertex with no edges. If a graph consists of a single vertex $v$, then the a path consisting of $v$ is vacuously Eulerian. It is also a Hamiltonian path, since it contains all of the vertices of the graph.