I don't think that there's any doubt that the technology exists. There's nothing clever about building an electrolyser to produce H2 & O2. There is a lot of cleverness in doing it efficiently, cheaply and in a compact form.
The Aquygen generator appears simply to be a H2/O2 mixture generator - again nothing special. This type of equipment has been used for decades for firing oxy-hydrogen torches. We had one of these torches back at high-school 15 years ago for doing fine metalwork/silverwork. The only difference was that the torch I used at school, enriched the H2/O2 mixture with methyl-ethyl ketone.
Oxy-hydrogen torches are old technology - as old, if not older than oxy-acetylene. So why are they so uncommon that noone knows about them. The answer is that they are crap. The combustion of hydrogen produces water vapour - and the water vapour withdraws considerable enthalpy from the flame in its latent heat of vaporization. Additionally, the flame is of very low density. So even though the flame temperature is very high (higher than OxyAc) the cutting/welding performance is terrible because the steam carries all the heat off.
People desperately trying to sell OH torches, show remarkable videos of how the flame is 'cold' but suddenly becomes 'hot' on contact with a material. It's an interesting property, but doesn't make the torch particularly useful. The higher density and low water content of an OA flame makes it far more effective for welding and cutting. Additionally, the high hydrogen content of an OH flame causes serious problems with welds - hydrogen is incorporated into the molten metal and causes serious embrittlement.
Most practical OH torches enrich the gas with MEK (as above) or ethanol/methanol - this adds carbon to the flame which drastically improves the flame density and reduces the heat loss through water vapour. It also reduces the point flame temperature which can be uncontrollably high in pure OH.
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What about the car?
It's long been recognised that hydrogen is an excellent fuel for ICEs. It burns quickly and cleanly and has a ridiculously high octane rating as well as a very wide range of mixtures at which it will burn. (Unlike gasoline, where a relatively precise mixture is required for decent performance).
However, building a hydrogen ICE is one thing - but hydrogen also benefits gas ICEs. Adding a trace of hydrogen to the air/gas mixture has a number of benefits.
1) The hydrogen increases the speed at which the flame propagates leading to faster and more even ignition of the mixture
2) Hydrogen is a very potent octane booster, allowing more aggressive timings for better performance and fuel economy and/or an engine design with higher compression ratio.
This is a well known property of hydrogen in ICEs. It is under active research by many university groups and all the major car manufacturers.
For one reason or other, most of the researchers are working on deriving the hydrogen from gasoline, rather than water. While I'm not entirely familar with the reasons why - some I've come across include:
1) An electrolysis system needs large quantities of high purity distilled water, as it will be contaminated by tap water
2) Electrolysis systems have traditionally been inefficient, bulky and needed large quantities of precious metals for the electrodes.
3) Direct reformation of gasoline for hydrogen is very efficient - considerably more so, than burning fuel to produce electricity to electrolyse water.