  We study superfluid properties of fermions on a 2D lattice using a
finite-range pairing interaction derivable from an extended Hubbard model. We
obtain signatures of unconventional pair-symmetry states, $d_{x^2-y^2}$ and
extended-s ($s^*$), in the BCS-BEC crossover region. The fermion momentum
distribution function, $v_k^2$, the ratio of the Bogoliubov coefficients,
$v_k/u_k$, and the Fourier transform of $v_k^2$ are among the properties that
are strikingly different for d- and $s^*$ symmetries in the crossover region.
Fermionic atoms in 2D optical lattices may provide a way to observe these
signatures. We discuss possible experimental ramifications of our results.
