  Using halo effective field theory (EFT), an expansion in $R_{core}/R_{halo}$,
where $R_{core}$ is the radius of the core and $R_{halo}$ the radius of the
halo nucleus, we calculate the charge and neutron form factors of the
two-neutron halo nuclei $^{11}$Li, $^{14}$Be, and $^{22}$C to
next-to-leading-order (NLO) by treating them as an effective three-body system.
From the form factors we extract the point charge and point matter radii,
inter-neutron distance, and neutron opening angle. Agreement is found with
existing experimental extractions. Results are given for the point charge and
point matter radii for arbitrary neutron core scattering effective range,
$\rho_{cn}$, that can be used for predictions once $\rho_{cn}$ is measured.
Estimates for $\rho_{cn}$ are also used to make NLO predictions. Finally, our
point charge radii are compared to other halo-EFT predictions, and setting the
core mass equal to the neutron mass our point charge radius is found to agree
with an analytical prediction in the unitary limit.
