After scaling and suitable Euclidean motions every rigid minimal patch can be placed on a unit catenoid of revolution $ x^2 + y^2 = c^2 \cosh^2 (z/c), c=1.$ with full area contact.
Is the statement correct? To me it appears to be so as principal curvatures $ |\kappa_1| = \kappa_2 = c/r^2 $ involve a single constant, so mean curvature vanishes.
EDIT1:
In such a case if $\psi$ is angle of reference direction to line of curvature, then for the constant $c$ we have
$$ c\, \kappa_n = \cos 2 \psi ; c\, \tau_g = \sin 2 \psi. $$