This paper introduces a novel hybrid high-order (HHO) method to approximate the eigenvalues of a symmetric compact differential operator. The HHO method combines two gradient reconstruction operators by means of a parameter $0<\alpha<1$ and introduces a novel cell-based stabilization operator weighted by a parameter $0<\beta<\infty$. Sufficient conditions on the parameters $\alpha$ and $\beta$ are identified leading to a guaranteed lower bound property for the discrete eigenvalues. Moreover optimal convergence rates are established. Numerical studies for the Dirichlet eigenvalue problem of the Laplacian provide evidence for the superiority of the new lower eigenvalue bounds compared to previously available bounds.