What is the problem if we try to interpret KG equation as a single-particle equation? Also, I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum mechanics.
There is no part of the physics involved that requires particle number to be conserved. Combine this with $E=mc^2$, which allows for energy to be converted into particle-antiparticle pairs, even when there is not enough energy to create such a pair, virtual particles allow for the temporary appearance of particles in a system that doesn't have sufficient energy for them. As such we cannot treat relativistic quantum equations as being based on particles, instead we treat them as fields with a certain energy, and assume the particle number of this field to not be constant.
If you try to construct the probablilty current for the KG equation, its zero component $j^0$ is not positive definite, even though you want to interpret it as a probability.