They probably do, the question is whether they can carry it to the site faster than the US navy, and whether the boron tanks have not been compromised.
"Wait, they need boron?"
Boron is a neutron absorber, and is therefore used to control the fission rate. You add boron when you're afraid of the fission rate getting out of control (e.g. Chernobyl). This is a legitimate concern here for two reasons:
1) A BWR is significantly more reactive (conducive to a high fission rate) at cold zero power than at hot full power, because cold water is a better neutron moderator than steam.
2) Xenon-135 and Samarium-149 are by-products of nuclear fission that have an effect on reactivity similar to boron. Their half-lives are on the order of hours, so when you crank down the fission rate, a couple hours later you also crank down the concentrations of Xe-135 and Sm-149, which, if you're not careful, can cause the reactor to go supercritical (and possibly prompt supercritical - a form of criticality in which everything happens approximately a thousand times faster - pretty much the worst-case reactivity excursion scenario) again a few hours after shutdown.
So, basically, you need enough negative reactivity from somewhere, either control rods or borated water, to counteract these two reactivity insertions. The Japanese reactor is almost certainly designed so that inserting all of the control rods into the core will kill any and all reactivity increases after shutdown.
What is most likely occurring is that there is very little nuclear fission inside the reactor right now. All the power inside the reactor is coming from decaying fission products, and it's probably on the order of kilowatts, it's just that the when the flow rate through the core drops from gallons per second to essentially nothing, a few kilowatts per cubic foot will get you pretty damn hot pretty damn quick. At this point some of the fuel rods have probably failed as well (if not melted), so the water in the reactor may be nastier than usual.
Unless the control rods have failed or are in the process of failing, I doubt that boron is even necessary for reactivity control, except as insurance. My educated guess is that their problems are entirely thermal- and containment-related, and that there is no danger of a reactivity accident, since a xenon transient or a condensation transient would have run its course by now, so the control rods probably have enough reactivity worth to keep the reactor subcritical indefinitely.
The heat inside the core is being produced by the decay of fission products, and there's absolutely nothing you can do to stop that except wait for enough half-lives that the activity slows down a bit.
Your general thesis, namely that as soon as they decide the reactor can't be salvaged, they'll dump in a bunch of boron just to be safe on the nuclear front, is correct, but where you're wrong is in assuming that adding a bunch of boron will help cool the reactor at all - it won't. All it will do is ensure that the reactor never goes critical again.
This is why spent fuel needs to sit in a pool of water for 5 years before anyone even considers moving it. Decay heat is serious shit, and it's not related to neutron physics at all.
EDIT: Probably time to get more specific about the term "reactivity". Reactivity is related to the "multiplication factor", which tells you how much bigger each generation of neutrons is than the last. It's zero when each subsequent generation is the same size - this is the normal operating state of a reactor, positive when each subsequent generation of neutrons is larger, and negative ... you get the idea.
If you've ever touched population dynamics in a differential equations class, you'll realize that this is a recipe for exponential growth and decay. Basically the time scale on which nuclear reactions proceed is "reactivity / mean neutron lifetime", with the caveat that if your reactivity is just above zero, neutron population growth is constrained by the longest-lived neutrons (it's like if every family has 2 kids, except for a couple hundredths of a percent, who have three, but put their third kids in cryostasis for a thousand years). In practice, this is how reactors are operated, because the mean neutron lifetime is _very_ small, so the worst thing that can happen is if your reactivity moves outside of this regime (this happened at Chernobyl). If your reactivity is negative, you are _always_ constrained by the longest-lived neutrons, but that's okay, because even this time scale is pretty short.
The upshot here is that if each successive generation of neutrons is smaller than the last, the number of neutrons (and hence things like fission rate that depend linearly on the number of neutrons) decays exponentially with a pretty short time constant.
The definition of "subcritical" is "having negative reactivity", so if the reactor is subcritical for any length of time, the fission rate will have exponentially decayed down to a tiny number.
So, no, given the state of relations between the countries and the general lack of experience in civilian application, not to mention the improbability of possessing the resources, I'd highly doubt that the US Navy would be instrumental in assisting the qualified and experienced staff at a nuclear power plant located in a highly developed country such as Japan. I think the author was just throwing that in there to add some ominous weight to his argument.