There's got to be a ground problem somewhere as #1 stated. When the horn is pressed it send a ground signal to from the horn button to the relay, I believe the washer pump is the same condition. Shared ground are canceling out the ground to the ignition circuit. Just saying it in different terms. Also if the battery light blinks, then there is something intermittent wrong with the charging system.
Is this consistent, or a random event? First thought would be approach this like any other engine stalling symptom, pull codes, and then monitor important data inputs while operating the systems associated to the problem. Advise get some testing done including pull codes and report back what they found.
There has to be a power or ground side fault occurring such that the operation of these other systems causes something essential to the engines operation to get shut down. An example would be if the radiator support is losing its ground connection to the rest of the body. When enough circuits use something like that for their ground path, operating one or two more things like the washers can cause enough current to flow that the voltage drop becomes excessive. I'd be scoping powers and grounds at the PCM, FICM, Grounds at engine block, inner fenders, radiator support. Power feeds to the ignition switch, battery cable splices etc. This would include using a scan tool and scope to monitor critical inputs to the PCM for injector control. Until some data is collected that starts to expose the fault, everything is in play and it takes patience and hard work to figure something like this out.
According to the schematic: The windshield washer and the horn both share the same ground connection, G108 at the right rear corner of the engine compartment. Other items that ground there are the right front headlight and right front marker lights, the blower resistor, the electric vacuum pump, but nothing that I would associate with any critical PCM inputs.
The horn relay contact however grounds through pin 91 grey/red of the PCM. That's a very odd thing for them to have done. Just for laughs, unplug the horn itself and see if hitting the horn contact in the column still produces the symptom and report back.