We present experimental results on the counting rate measurements for several single-gap $^{10}$B lined resistive plate chambers ($^{10}$B-RPCs) with anodes made from standard float glass, low resistivity glass and ceramic. The measurements were performed at the V17 monochromatic neutron beamline (3.35 .A) at the Helmholtz-Zentrum Berlin. For the $^{10}$B-RPCs with 0.28 mm thick float glass a maximum counting rate density of about $8times 10^{3}$ $Hz/cm^{2}$ was obtained. In the case of low resistivity glass and ceramic, the counting rate density did not deviate from linear dependence on the neutron flux up to the maximum flux available at this beamline and exceeded a value of $3times 10^{4}$ $Hz/cm^{2}$.