Like they wouldn't have boron near a nuclear reactor. I'm a PWR man myself, but I am 99% sure that GE BWR designs have a couple of big tanks of borated water specifically for a LOCA. I highly doubt they're having reactivity problems, it's probably just managing the decay heat, and now the radiation leaks, that's the problem. "Just" is a relative word, here, of course.
Boiling water reactors are simpler, cheaper, but generally aren’t made anymore because they are perceived as being less safe. That’s because the exotic coolant in the pressurized water reactor can contain boric acid which absorbs neutrons and can help (or totally) control the nuclear reaction. You can’t use boric acid or any other soluble boron-laced neutron absorbers in a boiling water reactor because doing so would contaminate both the cooling system and the environment.
He's completely wrong about industry adoption of BWRs. There are two BWR's planned to be built in the US (along with 3 or 4 PWRs), and I believe that China has contracted with GE for a few as well (along with 4 Westinghouse PWRs and maybe a few Areva ones too).
PWRs are preferred largely because of their higher power densities (a BWR core that produces the same power must be larger) and simpler nuclear calculations and control strategies (two-phase flow makes calculations much more difficult, and it's harder to calculate correct positions for control blades (whose effects are highly localized) than it is to calculate the correct boron concentration (whose effects are smeared over the whole core)). However, now that computers are faster and us nuclear engineers no longer have the excuse of slow computers to hide behind, PWRs are looking to move away from relying on Boron concentration as the main form of control (the Westinghouse AP1000, specifically, relies much more on rod movement than the AP600), because of the cost of performing regular boron dilutions.
He's right that BWRs are simpler and cheaper - about half the moving parts.