We apply the Landau-de Gennes theory to study the equilibrium problem that arises when a cylinder of radius R is kept at a given distance h from a plane wall. We assume that both the lateral boundary of the cylinder and the wall enforce homeotropic anchoring conditions on the liquid crystal, which prescribe the liquid crystal molecules to stick orthogonally to the bounding surfaces. Typically, in our study R ranges from a few to hundreds of biaxial coherence lengths, where a biaxial coherence length, which depends on the temperature, is a few nanometers. The equilibrium textures exhibit a bifurcation between a flat solution, where one eigenvector of the order tensor Q is everywhere parallel to the cylinder's axis, and an escape solution, where the eigenframe of Q flips out of the plane orthogonal to the cylinder's axis. The escape texture minimizes an appropriately renormalized energy functional F(*) for h>h(c), while the flat texture minimizes F(*) for h< h(c). We compute both the force and the torque transmitted to the cylinder by the surrounding liquid crystal and we find that the diagrams of both as functions of h fail to be monotonic along the escape texture. Thus, upon decreasing h, a snapping instability is predicted to occur, with an associated hysteresis loop in the force diagram, before h reaches h(c). Finally, since the symmetry of this problem makes it equivalent to the one where two parallel cylinders are separated by the distance 2h , the snapping instability predicted here should also be observed there.