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.
The other answer is incorrect. Breakers are standardized. The National Electrical Manufacturers Association (NEMA) is a good, honest group. Below is from NEMA
It is NEMA’s belief that standards play a vital part in the design, production, and distribution of products destined for both national and international commerce. Sound technical standards benefit the user, as well as the manufacturer, by improving safety, bringing about economies in product, eliminating misunderstandings between manufacturer and purchaser, and assisting the purchaser in selecting and obtaining the proper product for his particular need.
Hats off to the NEMA. It is great to find an organization that is honest.
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.