The quantum tunneling effect in the two-dimensional (2D) checkerboard lattice is investigated. By analyzing the pseudospin texture of the states in a 2D checkerboard lattice based on the low-energy effective Hamiltonian, we find that this system has a chiral symmetry with chirality equal to 2. Although compared to regular chiral fermions, its pseudospin orientation does not vary uniformly. This suggests that the perfect reflection chiral tunneling, also known as the anti-Klein tunneling, is expected to appear in checkerboard lattice as well. In order to verify the conjecture, we calculate the transmission probability and find that normally incident electron states can be perfectly reflected by the barrier with hole states inside, and vice versa. Furthermore, we also numerically calculate the tunneling conductance of the checkerboard nanotube using the recursive Green's function method. The results show that a perfect on-off ratio can be achieved by confining the energy of the incident states within a certain range. It also suggests that, by tuning the barrier, the checkerboard nanotube is able to work as a perfect ``band filter" or ``tunneling field effect transistor", which transmits electrons selectively with respect to the pseudospin of the incident electrons.