If you started a stream job (/v1/streamzip or /v1/streamzipid), the download link is not a direct S3 URL. The result endpoint returns a signed stream URL of the form:
https://api.s3zipper.com/stream?u=eyJhbGciOiJIUzI1NiIs...Open that URL (in a browser, or with curl -L -o out.zip "…") and the archive is streamed to you. The u token is what authorises the download, so the link works without an API token — share it carefully, and note that it expires.
For a regular zip job (/v1/zipstart), /v1/zipresult instead returns a presigned S3 URL that points directly at the archive.