Your (hydraulic-assisted) brakes will still work fine.
No, your (pneumatically-assisted) hydraulic brakes will suddenly not act the way you expect them to, because the engine is no longer providing vacuum to the pump.
You're both making bad assumptions. Brakes can be either of those things. Hydraulically-assisted brakes are commonly called "hydroboost" in the USA. They show up in places where either very high braking force is needed, as on F-Super Duty vehicles (starting at least as early as the '90s F250 Super Duty) or in places where there is no room for a traditional vacuum booster, as in the 2000+ Astro. But most of them are, as you say, vacuum driven. And the vacuum storage tank will give you one or two more activations after the engine stops in most cases, or in the case of hydroboost, the accumulator built into the system.
In either case, this is where I get to look smug, because my vehicles are simply not controlled in this fashion. I have mechanically-regulated diesels, a 1992 F250 (not Super Duty) and a 1982 MBZ 300SD. The latter actually uses vacuum to control engine fuel cut, but the former does it with a simple relay and without any electronics — just electrics. The truck does have a Kelsey-Hayes RWAL system (Rear Wheel Antilock Brakes) but that system cannot apply the brakes, it can only fail to apply them. Unfortunately, it's got a transmission with a computer. You would need the five speed manual to be completely impervious. In theory the transmission will work in 1st/3rd limp mode the control module fails, but if it does something very wrong it might be possible to destroy the transmission as well.