Can a binomial distribution have negative x values? The Poisson distribution makes a big deal that its great for modeling "Counts" of stuff (aka counts of monthly users) because its a non negative discrete distribution.
Well, aren't Binomial and Bernoulli distributions also non negative?
I can't think of any discrete distributions that can have negative x values. (Normal distribution i believe is a continuous distribution so it doesn't count)

Poisson, binomial, Bernoulli, negative binomial, etc. are just model distributions - that is distributions that are analytically tractable and/or can be derived under rather simple assumptions. One could thus reformulate the question as:

Are there known model discrete distributions with a support containing negative numbers?

Then the question becomes obviously about how we define these discrete distributions - the most well-known ones are defined with support on non-negative integers. Specifically the binomial has support
$$
k\in\{0,1,...,n\}$$
As @Tim pointed out in their answer, we can easily define a distribution with negative support, using the one with positive support.
This brings us to the phenomena that we actually describe: thinking of a problem where negative counts arise naturally could suggest a distribution (which is not necessarily a known and/or model one.) However, many real counts are by nature positive - like the mentioned counts of monthly users (unless we explicitly shift the origin, just as in @Tim answer.)
Remarks:

*

*Wikipedia has a list of discrete probability distributions, many of which are the distributions on integers. On a brief glance none of them has support extending to negative numbers.

*A similar question: Can the discrete variable be a negative number?
