Negative energy or work arises from comparing one statepoint to some reference.
Think of temperature scales, such as Centrigrade and Fahrenheit. They both have negative temperatures. But when we consider temperatures in the Kelvin and Rankine scales - with absolute zero as the reference - the temperatures are always positive.
Compared to absloute zero (correct me if I am wrong), everything has positive energy.
Negative energy is a 'relative' term.
One could ask is there such a thing as negative mass. As far as I know, mass is always positive.
Which reminds me of solving the diffusion equation for neutron flux in a reactor. One could select a function and boundary conditions that would give the possibility of a negative flux (or negative numbers of neutrons passing through a unit area per unit time). But a negative number of neutrons does not make sense physically - and a such a solution would clearly be wrong.