Neil is dying
If you pay close attention to Neil's sprite at the end when the red flash
happens, it looks like he takes painkilers... For what? Also, if you pay close attention to other little moments in the game, like when after Eva moves River from Johnny's memory
and Neil calls her a poser for visualizing a door after telling him she still had something to do, she wants to know why he's being so melodramatic and fighting so hard... and Neil doesn't respond... It hints that he's hiding something. Of course he could
just be addicted to painkillers...
- Also, if you select Neil to explore the house at the beginning of the game, when the fake RPG battle happens his HP is not at full like everyone else's, also hinting at illness.
The reason why Johnny is called Joey.
As stated by the game, Johnny and Joey were identical twins. Since Joey died in an accident and Johnny's memories were repressed intentionally, it's possible that their mother purposely called Johnny by both names to avoid him asking questions if people called him by the wrong twin's name. The story that the nickname was after his grandfather was likely a cover story in regards to this.
- Confirmed by Eva, although it's just a very educated guess on her part.
Eva is dying, and Neil is trying to reconstruct her memory, but not according to her wishes.
In the real world, Neil loved Eva, but she never returned his affections. So, when she decides to have Sigmund Corp. fulfill her wish, Neil takes the opportunity to relive some of their previous cases and alter things slightly to cast himself in a better light than Eva originally perceived. THAT is why Neil seemed to be so invested in Johnny's fate—to make himself seem like a more caring and sensitive person to Eva. Him sneaking off towards the end of the game had something to do with him managing the equipment hooked up to Eva. As for the stinger, the red flash was a sign of Eva's deteriorating condition, and the painkillers? Well, Neil's been working on the memories for several days now, and he needs to take something to be able to keep at it.
- I agree about the Eva is dying part - the part that convinced me was when Neil suddenly had to "feed ist digital pet" and ran off without explanation, presumably to take care of something in the real world or to alter some important detail of Eva's memory he had not thought of before. But I thought he had a different motivation: my guess is that this particular case was not solved, and in reality they never did manage to rework Johnny's memories to get him to the moon. This, maybe in combination with some other unsuccessful cases (which will be handled in the sequel(s)), filled Eva with guilt and selfdoubt, so Neil is in the process of reworking her memories so that she doesn't think of herself as a failure anymore and instead believes she helped all of her patients to die happy. This takes a lot of time and effort, which in turn is taking it's toll on Neil, who takes pills to keep himself awake. The red flash at the end was a sign of Eva's condition getting worse.