Helen, you can say ....on the following day or simply What happened the next day? I don't know that there is a grammatical explanation.

You don't normally use either! "What happened the following day." This is correct. It is much less common to add a preposition, but the preposition would still be "on".

I would say "on the next day" or "on the following day". Examples: On the following day, I went shopping. What happened on the next day? I cannot give you a grammatical reason for why this is so - only my native speaker insight.

Even though the preposition could be used I believe that in your case teaching students there you should omit it.

Hi, Jerry! I'm not sure about whether I should use preposition word here.