Why are protons positively charged, electrons negatively charged and neutrons have neither negative or positive charge?
Positive charge, negative charge or neutral is historically a matter of convention,This was classified in 1897 where the alpha particle,beta particle and gamma ray were discovered,in radioactive elements,all called rays at that time.
The classification of charge was according to the behavior of these rays or particles under the effect of the electric field,the positive and the negative charge terms were given according the deflection right or left,while neutral term was given to the ray that not affected by the field.So beta is a negative charge ,electron,and alpha is a positive charge,and gamma(photon) is neutral.
In 1911 the nucleus of the atom was discovered to be of positive charge,then it was made of proton(with +charge) and neutron a neutral particle.
This in brief,how the electron is with negative charge and the proton with positive charge and neutron is with zero charge.