Work your own schedule

I think I would say "work to your own schedule".

MustAskA job that allows you to work on your own schedule

I don't have a problem with that.

MustAskA job that allows you to work on your own schedule?

It's common in American English.

In the right context I would understand "on", but to me "work on your own schedule" most naturally means to work on creating or putting together your own schedule.

GPYmeans to work on creating or putting together your own schedule.

Yes, it can mean that, but without any other context, I would interpret it as a job where you have some latitude to adjust your working hours around your personal activities. This is often very useful for parents who have child or elder care responsibilities.