If you lay down a piece of cardboard, punch a central hole for the
squash plant, and then cover with mulch, brambles might make it
through, but most weeds won't.

When you ask about which plants are optimal for a no-till plot, I
assume you mean "clay plot". A sandy plot will be OK with any plant.
Plants that break the soil effectively include radicchio, fava,
cardoon, mache, and potato. Anything with a taproot, though the latter
two do not have one. Even carrot, parsnip, and beets, if you don't mind
the misshapen roots too much.