Special theory of relativity predicts negative energy as well $E^2 =p^2 + m^2$ which when merged with quantum mechanics predicts antiparticle (Dirac equation). But what does it mean that a free particle can have negative energy? I mean to say can I experimentally differentiate between a particle (say electron) with positive energy and one with negative energy? And also the general solution of Dirac equation is a linear combination of both positive energy (spin up and down) and negative energy (spin up and down) states, so what does it mean that if i measure the energy of an electron very large number of times, about half the time I will get positive and other half negative?
The existence of negative-energy particles is a common misconception.
In the context of relativistic quantum mechanics, it is possible to make sense of the negative energy solutions using the Dirac sea toy model. Imagine your particles to be fermions (that is, obeying the Fermi-Dirac statistics and consequently the Pauli exclusion principle). Moreover, imagine all of the negative-energy states to be filled. This state is called the Dirac sea (all negative-energy states filled and all positive-energy states unfilled).
The Dirac sea is proposed to model the vacuum state. It has an infinite negative energy, but that's no big deal, because we can redefine its energy to be zero (we can only measure energy differences). Consider the fluctuations of the Dirac sea. You can either have some of the positive-energy states filled (which would increase the energy), or you can have some of the negative-energy states freed (which would also increase the energy). As you can see, all fluctuations of the Dirac see have more energy than the sea itself, thus, there are no negative-energy particles!
The modern QFT-based treatment is a bit different.
In QFT you interpret positive-energy and negative-energy solutions as creation and annihilation operators acting on the Fock space of quantum states of second-quantized theory. The concept of "particle" is thus different from a "solution of the Dirac equation", however these are connected of course.
In QFT there are no negative-energy states, the vacuum having the lowest energy of all, which we usually redefine to be zero. All particles and antiparticles have positive energy. No Dirac sea is required.