Definition of Free field or Noninteracting field In QFT we can write a Hamiltonian operator for a free field. So, what is a free field/ noninteracting field?
 A: Let us speak specifically of electron. 
Definition of a free field is like a definition of a free particle: it is a particle in a rather weak or zero external field, so no essential trajectory deviation is foreseen for such a particle. Such an understanding is correct since in an external field you have a more general equation - that with the charge involved and with a more curved trajectory that can be verified experimentally. Factually, the particle equation in an external field serves as a model for a free particle. It has the same mass, charge, and spin.
A: A quantum field theory is defined by the connected n-point functions of its observables.  (One can recover the n-point functions from the connected ones, and from the n-point functions you can reconstruct the Hilbert space and all the operators.)
A field is free if its connected $n$-point functions vanish for $n > 2$.  Otherwise, it's interacting.  
Edit:  These are the basic observables of a QFT are the vacuum to vacuum matrix elements $\langle vac| \phi_1(x_1) ... \phi_n(x_n|vac\rangle$ of the products of the local observables $\phi_i(x_i)$ you get by evaluating a field $\phi_i$ at $x_i$.  These numbers can be assembled into distributions $G(x_1,...,x_n)$  on the product spacetime $(\mathbb{R}^{d+1})^{n}$, the n-point functions.  Intuitively, the n-point function's values represent amplitudes for having some number of field quanta appear or disappear at the $x_i$.
Connected $n$-point functions are certain combinations of the ordinary $n$-point functions that you get by subtracting out the $m$-point functions for $m \leq n$.  In the case where you only have one field, the simplest is $G_c(x_1,x_2) = G(x_1,x_2) - G(x_1)G(x_2)$.  You can recover the ordinary $n$-point functions from these, but they're a bit more useful physically.  Intuitively, they capture the situation where our field quanta appeared and disappeared at the $x_i$ by interacting with one another.
