Independent variables minimum counts for logistic regression

"Univariable" logistic regression (which I take to mean using only a single predictor for a binary outcome) is seldom wise if you also have information on other predictors, because of the inherent omitted-variable bias in logistic regression. If you omit any predictor associated with outcome, whether or not it's correlated with the included predictor, your estimate of the coefficient of the included predictor will be biased.

For standard "multivariable" (or "multiple") logistic regression you should generally have on the order of 15 members of the smaller outcome class per predictor that you are evaluating. If one of your predictors is binary and has few cases in one of its classes then the estimate of its coefficient will probably be imprecise, with large standard errors. Depending on the overall structure of the data you might have troubles fitting the model, for example finding perfect separation, but I don't know that there's a general rule for class prevalence within a binary predictor.