  The Glashow-Iliopoulos-Maiani mechanism is extremely efficient to suppress
the flavour-changing neutral current decays of charmed hadrons induced by the
$c \to u$ transitions, making such processes particularly sensitive to
phenomena beyond the Standard Model. In particular, $c \to u$ decays with a
neutrino pair in the final state are theoretically appealing due to the small
long-distance contributions. Moreover, in the framework of the Standard Model
Effective Field Theory (SMEFT), the $SU(2)_L$ invariance allows to relate the
Wilson coefficients in the effective Hamiltonian governing the $c \to u \nu
{\bar \nu}$ decays to the coefficients in the $c \to u \ell^+ \ell^-$
Hamiltonian. We analyze the $B_c \to B^{(*)+} \nu {\bar \nu}$ decays, for which
branching fractions of at most ${\cal O}(10^{-16})$ are predicted in the
Standard Model including short- and long-distance contributions, so small that
they can be considered as null tests. Using SMEFT and the relation to the $c
\to u \ell^+ \ell^-$ processes we study the largest enhancement achievable in
generic new physics scenarios. Then we focus on a particular extension of the
Standard Model, the 331 model. SMEFT relations and the connection with $c \to u
\ell^+ \ell^-$ imply that ${\cal B}(B_c \to B^{(*)+} \nu {\bar \nu})$ could
even reach ${\cal O}(10^{-6})$, an extremely large enhancement. A less
pronounced effect is found in the 331 model, with ${\cal O}(10^{-11})$
predicted branching fractions. Within the 331 model correlations exist among
the $B_c \to B^{(*)+}\nu \bar \nu$ and $K\to \pi \nu \bar \nu$, $B\to (X_s, K,
K^*) \nu \bar \nu$ channels.
