  Although lattice gases composed by $k$NN particles, forbidding up to their
$k$th nearest neighbors of being occupied, have been widely investigated in
literature, the location and the universality class of the fluid-columnar
transition in the 2NN model on the square lattice is still a topic of debate.
Here, we present grand-canonical solutions of this model on Husimi lattices
built with diagonal square lattices, with $2L(L+1)$ sites, for $L \leqslant 7$.
The systematic sequence of mean-field solutions confirms the existence of a
continuous transition in this system and extrapolations of the critical
chemical potential $\mu_{2,c}(L)$ and particle density $\rho_{2,c}(L)$ to $L
\rightarrow \infty$ yield estimates of these quantities in close agreement with
previous results for the 2NN model on the square lattice. To confirm the
reliability of this approach we employ it also for the 1NN model, where very
accurate estimates for the critical parameters $\mu_{1,c}$ and $\rho_{1,c}$ --
for the fluid-solid transition in this model on the square lattice -- are found
from extrapolations of data for $L \leqslant 6$. The non-classical critical
exponents for these transitions are investigated through the coherent anomaly
method (CAM), which in the 1NN case yields $\beta$ and $\nu$ differing by at
most 6\% from the expected Ising exponents. For the 2NN model, the CAM analysis
is somewhat inconclusive, because the exponents sensibly depend on the value of
$\mu_{2,c}$ used to calculate them. Notwithstanding, our results suggest that
$\beta$ and $\nu$ are considerably larger than the Ashkin-Teller exponents
reported in numerical studies of the 2NN system.
