Do independent variables need to be normally distributed in negative binomial regression? From searching around, I see that independent variables in logistic and linear regression do not need to be normally distributed, but I haven't found any information about negative binomial regression.
Is there any need for independent variables to be normally distributed in negative binomial regressions?

NO
This is a common misconception about generalized linear models. I think people see that the error term in an OLS linear regression is supposed to be normal (for the usual inferences to work) and take that to mean either the pooled distribution of all $y$ values must be normal or that the features must be normal, both of which are false.
In particular, binary $0/1$ variables are perfectly okay in generalized linear models (such as a GLM with a negative binomial likelihood).
