Overcoming our vertebrate bias
There is no particular reason we should expect real world biology to apply to rulings like this; D&D is not a Biology simulator, and this is a game where for the purposes of spells mushrooms count as plants and scorpions count as insects.
However, if your DM does wish to consider real world biology, I would say that most of a snail's shell is a dead mineral, calcium carbonate. Only the thinnest outer lip of the shell is alive, the part that is continually added to by an organ called the mantle. For our mammalian bodies, the hair and fingernails would be good analogies, since they are (keratin-based) dead materials that have a living edge connecting to our body where they are grown. This is why you can cut your hair or clip your nails without damaging yourself; those parts of your 'body' are already dead. This is not a perfect analogy, though, since as Mołot points out, a snail's shell encloses and protects its living organs - much the way our skull encloses our brain, but our bones are much more alive than a snail's shell is.
So, if something had ripped out my fingernails including the nail bed, or tore out my hair including the skin cells of my scalp, what would the regenerate spell bring back - just the scalp and nail bed, or the full-length hair and nails as well?
BBeast, in their answer to Does hair count as “body parts” for the Resurrection spell?, points out that Dream, Scrying, and Simulacrum all explicitly include both hair and nails as "body parts". This is more about western traditions of sympathetic magic than real world biology, but there is clear precedent within the D&D spell rules for saying that hair and nails count as body parts. If they count, then in this DM's opinion, snail shells should also count for Regenerate.