The authors consider the neutral equation \[ [x(t)-x(t-\tau)]^{(n)}+Q(t)x(t-\sigma)=0, \qquad t\geq t_0,\tag{1} \] where \(\tau=\text{const}>0\), \(\sigma=\text{const}\geq0\), \(n\in\mathbb{N}\) odd, \(Q\in C([t_0,\infty),{\mathbb{R}}^+)\) and is not identically zero eventually. Let \(M_n=\max\{x(1-x)\cdots(n-1-x)\colon x\in[0,1]\}\). It is proved that if \[ \liminf_{t\to\infty}t^n\int_{t}^{\infty}Q(s) ds> {M_{n+1}\over n}\tau, \] then any solution of (1) oscillates, and if there exists \(T\geq t_0\) such that \[ t^n\int_{t}^{\infty}Q(s) ds\leq {M_{n+1}\over n}\tau,\qquad t\geq T, \] then the equation (1) has at least one eventually positive solution. It is also proved that any solution of (1) oscillates if there exists a function \(\varphi\in C([0,\infty),[0,1])\) such that \[ \int_{t_0}^{\infty}[1-\varphi(t)]\overline{Q}(t) \exp\biggl[2\int_{t_1}^{t} \bigl[\varphi(s)\overline{Q}(s)\bigr]^{1/2}ds\biggr]dt=\infty, \] where \[ \overline{Q}(t)=\begin{cases} \displaystyle{{1\over(n-2)!\tau}\int_{t}^{\infty} (s-t)^{n-2}Q(s+\sigma) ds,} & n>2,\\ Q(t), & n=1\end{cases}. \]