It's stay AT home
I didn't understand this. It sounds like this is a grammar objection, but "stay home" sounds find to me. I googled it. Apparently "stay home" is fine, but it's a bit of an exception to the general rules. Something to do with "home" being both a preposition and a noun.
(Further googling points out that "stay at home" and "stay home" don't always have the same meaning, it depends on the context. Someone attending a conference in their own city might tell a colleague they were going to "stay at home", rather than at a hotel, but if they said they were going to "stay home", that would imply they weren't going to leave home for a few hours to attend the conference.)