Does an outdoor hot tub receptacle and breaker both have to be GFCI?
GFCI is not a receptacle or breaker.   It is a zone of protection provided by the GFCI device.  It protects everything down the line from the GFCI device.  

Sometimes the GFCI device itself has a couple of convenience outlets that are on the protected side: a GFCI+receptacle combo device.  This is what you mean when you casually say "GFCI", because this is the most visible form of GFCI device. 

You are also familiar with the GFCI+circuit breaker combo device. This obviously protects loads downline, since it has no convenience outlets of its own, just two "LOAD" terminals.  GFCI+breaker combos are $40-ish. 

Then there is the GFCI-only (not a combo) device, a "black box" which does only the GFCI function.  It has two LINE inputs, two LOAD outputs, and that's all.  It typically comes in a rather familiar looking package, hence the slang term "deadfront".  Deadfronts are about the price of the cheapest GFCI+receptacles since they don't need child-safe sockets nor outdoor rating, around $15-18. 

 

Since this is a dedicated circuit for 1 appliance, you also need a single plug. Common and $5 in plain sockets - rare and expensive in GFCIs. 

Why do we care where we protect? 


First, are the wires protected?  If the GFCI+receptacle gets wet, or water gets inside the box, water can contact the non-protected wires directly. The GFCI tripping won't help you at that point. Not an issue with GFCI+receptacles upstream. 
A GFCI outdoors is also ravaged by the elements and will have a much shorter service life, which is a shame because it also must be an "outdoor-rated" GFCI at extra cost.
Anti-tamper safety doors over the blades are required, these are much cheaper on plain receptacles than GFCI receptacles. 
Cost -- GFCI+breaker combo devices are around $40 and you are married to your panel brand*; if your panel is an obsolete like Pushmatic, you're out of luck.   Deadfronts cost $15-18 + $5 for a plain breaker.  Outdoor rated, anti-tamper GFCI+receptacles cost $25+. 




* Siemens makes breakers specifically for Square D QO panels; and UL lists them for QO panels only (they don't fit Siemens panels). However they are not any cheaper than Square D's QO breakers, so these "classified" breakers are a waste of time. So Don't mix and match brands, even if they "seem to" fit.