Is it possible to breed neanderthals through selective breeding? I've heard most non-subsaharan africans have neanderthal DNA with it being more prevalent in northern regions, that sometimes 1-4% of the DNA has neanderthal origins. Speaking strictly scientifically, would it be possible through selective breeding to breed an individual where the majority of the DNA is neanderthal?

No, it is not. As you said only 1-4% of non-subsaharan africans' genome is from Neanderthal and it is more or less the same sequences. The entire Neanderthal genome is not present in modern day humans, it is only a small set of sequences.