As I understood, the Higgs field has a non-zero vacuum expectation value (vev) on zero energy.
For example, in the case of the non-quantized electric field, it would be measured in $\dfrac{J}{C}$, i.e. $\dfrac{\textrm{energy}}{\textrm{charge}}$. Thus, it shows the energy in the field per unit charge.
How could it be a simply energy-like value in the case of the Higgs field?
When we say field, you shouldn't immediately think electric field. You should instead think of the easier scalar field. Also the electric field is not J/C. That is the voltage potential function. Now look at how the action of the field configuration depends on the field you are looking at. For example, it might be $S=\int d^4 x \partial^\mu \phi \partial_\mu \phi$. We want this to have dimensions that an action needs which is the same as the dimensions of the constant $\hbar$. Now do a dimension count. Usually people ignore $c$ and $\hbar$ so that they can just count powers of $GeV$ and then use $c$ and $\hbar$ to fix it later.
