I have a cutler hammer breaker box with a cutler hammer breaker that will be fed to a disconnect box with a GFCI before going to the hot tub. Does the disconnect box necessarily have to be a cutler hammer brand with cutler hammer breakers. I wouldn't think so but want to be sure.
There is no code in the NEC that requires you to use matching breaker to the panel box. Generally there are some that are universal. i.e. Square D Homeline, GE, many Siemens, Murry, ITE and others. A few are brand specific...i.e. Square D QO series, Cuttler Hammer Tan Handles. But you may certainly use an ITE breaker in a Murry panel...and so on.
But to answer your question. No you can use any brand disconnect box that you so desire.