Logistic regression correcting for twin data

It would more like probit than logistic regression, but you could set up your analysis as an ordinal-threshold model. You'd set up your dataset to have one row per pair of twins. You'd have separate groups (submodels) for MZ and DZ twins, and allow the twin correlation to be different for MZs and DZs. The variance of the latent continuum would be fixed to 1. You'd need to fix one of either the latent mean or the threshold to zero. You'd use definition variables to condition the value of the other parameter, for each twin in each row of the dataset, upon his/her values of the predictors. That is, the value of the other parameter for twin j in twin pair i would be b0 + b1*x1ij + b2*x2ij + ..., where b0 is a regression intercept, x1ij is twin ij's score on covariate #1, b1 is the regression slope for covariate #1, and so forth. The covariates would be definition variables, and the regression coefficients would be free parameters, constrained equal across zygosity groups.

All together, this would model (1) the dependence of twins within pairs, (2) the greater degree of within-pair resemblance for MZs compared to DZs (which will be so for any trait having nonzero heritability), and (3) the regression of a dichotomous phenotype onto one or more predictors. But, be advised that the sign of the regression coefficients will depend on whether they predict the threshold or the mean, and that the coefficients don't have the same interpretation as those from logistic regression.