Hi Christy,
pardon my French,
but that is about the most half ass'ed way I have ever heard of
fixing cracks in a slab before doing tile. I would find another tile installer.
by any chance are you in Florida ?
your best way to do it is to remove the old vinyl and install either a
Noble - CIS,
Custom Building Products - Fracture Free or a Anti-Fracture Membrane like
ProtectoWrap.
there is no cheap way around it like this installer is telling you.
hopefully he wont tell you about using tar paper.
(edit): I see from your other threads your are in Baton Rouge.
is this still the same floor you were doing Wood looking plank tile on ?