You take a company down, and automated sourcing puts them back.
Block company from Grow → Backfill → Company sources, or Block from backfill on the Companies row menu. Published jobs for that employer are archived. Backfill, CSV and XML import, and API bulk writes reject them. The company stays on your board so you can unblock later. Unblocking does not republish the archived jobs.
You can still post a job for them yourself, and employers can still post if that is enabled on your board.
See Company blocklist.