The Father sent the Son. The Son came, lived, worked miracles, and sacrificed his life for the joy set before Him. I see the joy set before Him as the multitude who become saved. Is there Biblical evidence that he would have died just for me, even if I were the only one ever to be saved?
Like you, I have heard the preaching point a number of times that "even if you were the only person in the entire world who sinned, Jesus would have died for you." Obviously, there is no scripture which says this. The Bible does not deal with abstract questions about realities that do not even exist. Anyone saying this is drawing a conclusion from his/her knowlege of God from the Bible. It is really speculation/extrapolation based on the unimaginable love and grace of God.
It is not really very useful to make conjectures about unrealistic scenarios. You would probably do well to not worry about the literal truth or lack of truth of the "even if I were the only sinner…" question. I believe those who say this are sincerely trying to convey the love of God for each of us. However, I would not take this claim literally. Probably the preacher making this point is not even trying to speak literally, but to convey an idea about how great God’s love for each of us is.
There is one scene in Genesis which may shed at least a tiny bit of light on your question. It is the scene in Genesis when Abraham asks God to spare Sodom for the sake of the righteous people in the city. The situation is only weakly analogous to your question, but perhaps worth looking at. The story is found in Genesis 18:22-33. Abraham asks God to spare Sodom if there are even just 50 righteous persons in the city. God agrees to the request. He then argues God down to sparing the city even if there are just ten righteous people in the city. God agrees to the "bargain." But the request stops there. Would he have spared Sodom of there were just one righteous person in the city? Apparently not, because we know from 2 Peter chapter 2 that God considered Lot a righteous man, he was in the city, and the city was not spared. I would be very cautious to apply this story to your question, but thought it might be food for thought.
In conclusion, God loves each and every one of us very much. He does not want anyone to perish, but wants all to come to repentance and salvation (paraphrasing from 1 Timothy 2). Whether he would come down and die on a cross if there were even just one person who needed to be saved is not a real question about a real situation, so I will choose not to speculate about this question. I guess that in that situation, the one sinner would have to be the one who killed Jesus! Not a good plan. The reality is that he did come and die and you can be saved by the blood of Jesus. Hopefully, this will be good enough for you.