You have to send user as an array. So /api/public/signup?user[user]=username&user[email]=... should work.
See also: https://github.com/agentejo/cockpit/blob/next/modules/Cockpit/Controller/RestApi.php#L65
And you shouldn’t send passwords as get parameters. They might pop up in log files, browser histories or analytics data.
And please provide some more info when asking for help. I had to read your previous post to understand, what you are doing.