No, he would not - because in the past of the "now" timeline, Reese appearing has happened. It can't be (easily) undone.
According to the Sarah Connor Chronicles, time travel in the Terminator universe works along some variation of the many-worlds interpretation (and is implicitly supported by the fact that John still exists as of T-3, even though that's a different timeline1).
However, it's a very odd variant that suggests time travel into the past is actually pretty restricted by the in-universe laws of physics - you can't go to whenever you want to, which would explain why only a couple of Terminators went after the Connors.
The following is theory based around what shows up in the 4 movies and the TV series.
- Prior to T-1, there was a "most likely future". This is where Reese and the Terminator came from.
- Time travel from possible futures that occur when the "present" is after this point cannot alter this event. You can consider it similar to Doctor Who's concept of "fixed points in time".
- Due to the events of T-1, the "most likely future" between T-1 and T-2 is now something different. As I mentioned in my other answer, the Terminator from T-2 did not come from the same Skynet as the one in T-1.
- The events of T-1 after Reese's arrival have already changed the future. By the logic of the question, he and John should already have blinked out of existence.
- The events of the "most likely future" prior to T-1, the future that Reese was from, still happened - it's just that, that particular section of the timeline is inaccessible from the current "present".
- Timelines continue shifting around like that between T-2 and either T-3 or TSCC, depending on which route through the timelines you want to take
The above theory is supported by TSCC where, at one point, we have two people currently in the present that had time-traveled from different futures. Their memories (including the date of J-Day, if I recall) didn't match up.
On the other hand, there is one point in TSCC that don't quite jive with what I've described here:
- There was a terminator that arrived nearly a century too far in the past. His target wasn't any of the Connors, but a politician. He became a businessman, built a building, went into suspend mode inside that building, and waited for the politician to be born and have his life on-track to be in that building at a specific date and time, so that the Terminator could assassinate him.
- This seems to only work if the timeline did not shift drastically upon that Terminator's arrival. However, given the extreme time span between his arrival and the change he intended to make, that could be enough time for events to smooth out. Cameron also put a stop to that plan before it could be carried out, also possibly preventing a timeline split.
1 For why we know it's a different timeline, see this post.
Finally, and put last because it's less fun, we have Word of God:
- Anton Yelchin talks about his character, Kyle Reese's origins, and the original timeline he came from in The Terminator. "When Connor sent Kyle back, that was a world in which Kyle wasn't Connor's father. So when he sent him back, it then started this chain of the Connor that you have in [all the sequels] where Kyle Reese is his father — it'll be interesting how they tackle that [in future sequels] if we ever get to a point we have to send [Kyle] back."
- McG states that Terminator Salvation uses "... the spirit of parallel worlds, as theorized by Einstein. We try to pay attention to that approach to a fundamentally theoretical construct."3 This allows Skynet to be aware of the other attempts on the life of John Connor, and allows it make plans based on it's past successes and failures. This likely explains why Skynet would lure John into combat against the T-800 Model 101, because it is the model that managed to successfully kill him in a different future on July 4, 2032.
- McG goes on to discuss what would happen if Kyle were killed in Terminator Salvation, and the repercussions it would have on the timeline. "Will he be erased in the photograph, like in “Back to the Future”? That’s an excellent question that theorists have been bandying about for the ages. We play it more simply. Kyle Reese must be kept alive, so he can be sent back in time from 2029 to protect Sarah Connor, impregnate her and she’ll give birth to John Connor who will save us all. And the simplest way to understand that is to protect the triangle of Kyle, John, and Sarah. Any deconstruction of that leads to more headache than satisfaction."
Note that the 2nd bullet point I have copied here (it's the 3rd on the wikia page) indicates that the different versions of Skynet from the future are interacting with each other - the newer versions are learning from the Terminators left in the past by the older versions. Further evidence that neither Reese nor John would simply disappear.