How would you install commercial 2X4 ceiling tile to a dry wall ceiling?

Sheet rock screws.

Alternative:

If you have a hole in your ceiling and want to fix it you use a piece of wood to hold a piece of Sheetrock which you will finish with plaster or joint compound.

You would cut a piece of wood more narrow than the hole and longer than the hole.

Put the wood into the hole so it cannot fall out. Cut a piece of Sheetrock to about match the hole and screw the new piece of Sheetrock to the wood.

Use joint compound to give the repair a finished look. If you simply put joint compound into the hole it will crack and take many coats.

Paint.