As for the law about guns on school property, it does not matter if it is a private or public school.
Only the OP's location gives a clue that he's asking about Mass laws, as it is posted in General not in the Mass Laws sub-forum.
If you read MGL the definition of a school (in MA, Fed definition is different of course) includes only elementary, secondary and college level schools. Anything co-located there (e.g. pre-school, kindergarten, etc.) would be off-limits.
So pre-schools are not off-limits per MGL.
On the other hand when you go to get a Mass Dealer's License anywhere near a school, pre-school, church, etc. beyond the law, you are at the whim of your local police chief and town fathers/mothers (selectmen, town council, planning board, zoning board, etc.) and if they don't like it you will have a very long and expensive battle on your hands.