Short version: Maybe, but the scientific literature is really bad for this. It also seems to depend on how long it takes to extract juice from a certain amount of fruit. The longer it takes, the more likely it has some impact.
First, do different juicing methods affect anything about the juice they produce? There doesn't seem to be a literature on the effects of different juicers on nutrient content, but there is a modest one on the effect on flavours, juice yield, and amount of particulate in the juice. The main difference seems to be that cold pressing ends up extracting juice from only the flesh of the fruits, while "hot pressing + maceration" (which is an industrial process that seems similar to a blender or centrifugal method) extracts from seeds and/or skins as well. If there are nutrients that break down when heated, and are present in the flesh of the fruit, it stands to reason that cold pressing would recover them better. This paper has a good discussion of the effects in grapes, but doesn't address nutrition explicitly, only certain oils that affect the flavour.
This rather interesting book covers (starting around page 50), the different things that can affect flavour, colour, clarity, etc. of juice at small and medium production scales. Using a hot press (where the fruits are heated to ~60 degrees and possibly macerated before juicing) does have some effects.
Okay, so what sorts of nutrients might break down during hot pressing? This paper shows that at least vitamin C (which is presumably a nutrient of interest in juicing) is prone to degradation at the temperature range used in hot pressing (~60 C). However, they exposed broccoli to 60 C temperatures for 10 minutes continuously to get that kind of effect, so it's unclear whether exposure during juicing would be long enough.
Can a home juicer get that hot? While, a centrifugal juicer is basically a blender. I could not find any good information on the temperatures they reach (just a lot of unsourced health websites). However, there are definitely higher end blenders capable of bringing their contents to boiling or near boiling temperatures. This Cooking SE question covers how to make soup using only a blender (heat by running on high), and this blender website advertises that you may be able to boil soups in some of its models. It stands to reason then that a lower-end blender should also heat its contents, even if it doesn't go quite as fast. Since it only needs to get up to 50 or 60 C to damage e.g. vitamin C, it might be able to do that.
However, this paper, appears to show that pomegranates juiced via blender actually yield more citric acid than those juiced by press (tables 1b and 1c). While citric acid is not a nutrient, this shows that some compounds are better extracted via blender than press, so even if say, vitamin C, extraction were reduced, some other nutrients might be present in higher amounts (maybe essential oils from the seeds?).
So overall, what can be said is that:
There doesn't seem to be a real, public, study that looks at these effects directly. The juicing industry (and therefore juice science) is very concerned about how your juice looks, tastes, and smells, but doesn't seem to care about whether its nutrition is affected by the extraction method.
There is a plausible mechanism of action for a centrifugal juicer to degrade nutrients: spinning stuff fast and hitting it over and over makes it hot, and hot stuff tends to degrade certain types of nutrients (though if it gets too hot, the nutrients can become stable again).
There is also a plausible mechanism of action for a centrifugal juicer to get more nutrients: spinning stuff fast and hitting it over and over can extract things from more parts of the fruit than cold pressing it can.
Just a maybe. It does seem like in general, cold pressed juice has better "flavour" properties, but hot pressing yields more actual juice though. So maybe that can help you decide what to buy.