Here's why it shouldn't work. Read up on the efficiency numbers.
Wikipedia: Electrolysis of water
It is hard to believe that mileage would benefit if the conversion of water -> electrolysis -> Brown's Gas -> burning -> heat were 100% efficient. At 33% efficiency which looks in line with the loss of electrolysis and the heat loss of an engine we can see there an immediate loss of energy to be overcome if there is to be any increase in mileage. Since there's no obvious reason why the introduction of Brown's gas, hydrogen, or oxygen in the fuel air stream should substantially increase the efficiency of an engine which many claim is already close to maximum efficiency, science says that a Brown's gas generator powered by the alternator should produce more loss than gain.
The system consumes distilled water, electrolyte, and electrodes, none of which are free, though the claim is that 316L stainless corrodes so slowly that it's cost should never be an issue.
They have an EFIE like device. Hacking into any of the emissions sensors is illegal without EPA approval. You need an EFIE because lots of extra oxygen shows up in the exhaust. Since the hydrogen and oxygen are perfectly matched, there must be extra hydrogen there too, yet more waste and not good for your catalytic converter.
Fuel and air are carefully regulated. Depending on how you hook these up and how well you maintain the conductivity of the electrolyte the Brown's gas stream is essentially unregulated, or possibly regulated to the opposite of what you want. It's hard to see how this would improve anything without the ability to produce dependable quantities of gas matched to engine operating conditions.
The technical writing on all these HHO sites is poor and fuelfromh2o is worse than average which detracts from their credibility already ailing from pushing something expensive that science and industry says doesn't work.
I say it's a horse race because there are enough good claims about Brown's gas generators that they can't all be deluded, lying, or incapable of measuring correctly. Even if they do work, they may not work on your vehicle or you may be unable to make one work properly. That's why you need to secure your ability to return it before buying.
I know someone that put a 4 cylinder Dutchman HAFC kit onto a V8 and the mileage went from 18 to 26. The trouble is they put the entire kit in which includes fuel heat, an EFIE equivalent O2, and gas additives and the vehicle is too old to be tested with a Scanguage so they have no idea which, if any, caused the increase and tank mileage testing makes it impractical to remove various things for a short time to see how the mileage changes. They paid, put it all on, and got enough gain to avoid exercising the return policy. Could they have got the same benefit from low cost products or better maintenance? Hard to tell!
HHO generators need to get a lot cheaper or better accepted before I'm willing to put my money on it.