The only time you have to use a Priority Mail-branded box is if you're shipping a Flat Rate box or envelope. In that case, your postage has to be for the same type of Flat Rate packaging that you used. For example, you have to use a Priority Mail Small Flat Rate Box® if you purchase Priority Mail Small Flat Rate Box® postage.
Otherwise, Priority Mail® is just the name of the service class, and you can use any box or packaging that you'd like... and for most cases, it's actually cheaper to use your own packaging instead of a Flat Rate service.
Use our postage rate calculator to find your most cost effective options.