You can look the definition of a virtual particle in any text-book or jld:s answer, since it is just a definition.
About existence of anything, it becomes immediately an ontological problem and is subject to interpretation. On the advent of quantum mechanics, its opponents demanded reality. In 2016, reality of photons or virtual particles are sought after in StackExchange Physics.
I'd say virtual particles are as real as real particles, to the extent that it makes sense to talk about reality of physical objects.
There are a few complictions however in the matter.
It is very hard to define particles with interactions. Given an electronic system with some degrees of freedom and photon system with some more degrees of freedom, the quantum evolution makes all exact single particle pictures vanish/invalid. For example removing an electron from an atom takes 5eV of energy, that energy consisted of huge amount of virtual photon interactions etc. It was not the energy of the electron only. Particles are just approximate tools to understand quantum fields anyway. That is why all the confusion in 'What is really photon?' question and all the confusion about particle-wave dualities and such. I believe Higgs is just a resonance of a field and always virtual since it is not observed directly. But nobody says that Higgs is not real (neither do I).
Particles propagating asymptiotially free are easier to define, since they no longer interact. Poles of their propagators are point like, giving well defined energy and momentum for example. The whole scattering theory is based on these asymptotic free particle states and whatever interactions may occur while scattering are called virtual.
Something being hard does not make it real/virtual I don't quite buy most of the arguments about virtual particles not being real. They follow the same equations of motion (they "are"/are described by the same propagator) , so what if they are not in their mass-shell. If something behaves quantum mechanically and has due to interactions lost its single particle picture, that does not make it less real than the special asymptoic solution of the particle in free space.
And some more loose discussion
There is always "quantum state relativity" (I just invented that word) in quantum mechanics. If you consider detector or you being inside of your scattering matrix, then you become a virtual collection of propagators yourself. So what is inside and outside of a quantum system has to be defined. (Usually it is trivial in a normal experiment.) Now we define that what happends inside is called virtual and what is coming/going from/to outside is called real because we as humans still keep insisting on classical reality! Just like in 20's.