Do you have to get a straight talk cell phone with the straight talk plan?

Technically, yes. While Straight Talk is a pay as you go cell phone service, buying minutes is referred to as buying a "plan." (Plans last last between 30 and 365 days, depending on which plan you choose)

If you want to ensure that your service is never interrupted between plans, Straight Talk customers can set up "Auto-Refills" - a rescheduled automatic payment to refill your plan on your service end date.