How can I interpret log-transformed independent variables in terms of percent change in a negative binomial regression?
Interpretation depends on the link function.
i) if the (typical) log-link is used then the coefficients relate log-change in y to log-change in x, and so the interpretation would be the same as for a log-log regression -- roughly speaking they represent "percentage change in y for a 1% change in x" (as long as the coefficient isn't large). However, on the link scale, we wouldn't be talk about the coefficient giving expected percentage change.
ii) if some other link is used, some other interpretation would be needed -- but again, the interpretation is again similar to in the regression case; for example, an identity link would yield a similar coefficient-interpretation as for an ordinary regression where $x$ is logged but $y$ is not.