Is it appropriate to do a logistic regression where both the dependent and independent variables are binary? for example the dependent variable is 0 and 1 and the predictors are contrast coded variables -1 and 1 ?
There is no reason not to do this, but two cautionary thoughts:
Keep careful track during the analysis of which is which. In large projects, it can be easy to get lost, and produce errant results.
If you choose to report regression estimates, rather than odds ratios, make your coding scheme clear in your report, so readers don't produce inaccurate ORs on their own assuming they were both coded 0,1.
May seem basic, but I've seen both problems make it into published papers.
For, clarity: the term "binary" is usually reserved to 1 vs 0 coding only. More general word suitable for any 2-value coding is "dichotomous". Dichotomous predictors are of course welcome to logistic regression, like to linear regression, and, because they have only 2 values, it makes no difference whether to input them as factors or as covariates.
Typically it helps interpretation if you code your predictors 0-1, but apart from that (and noting that it is not required), there is nothing wrong with this. There are some other (contingency-table based) approaches, but if I recall correctly, these turn out to be equivalent to (some form of) logistic regression.
So in short: I see no reason not to do this.