If I have a personal checking account with a zero balance for some time, will the bank eventually close it automatically?

At my bank, yes. But not all banks will. The bank you have your account at is the only one that can really answer this question.

At our bank it would automatically close too.

If there is a monthly servicing fee, you may wish to close it yourself as some financial institutions will keep them open, keep charging your monthly fee and over draw your account then charge you at a later time. It happened to one of our customers at another bank and it is perfectly legal for them to do so.