I am hypothesizing a 1-1-1 mediation model where X --> M --> Y. M and Y are binary. I am interested in whether this process varies by characteristics of level-2 units (cross-level interaction). Can this be done in Mplus? Can anyone provide citations for resources on this type of analysis?
Yes, Mplus can do this. You may want to use either 2-level WLSMV or 2-level Bayes in order to access M* and Y* continuous latent response variables that make the X->M*->Y* relationships linear and indirect effects defined as usual. Look for references by Preacher and Zyphur, e.g. Preacher et al (2011) in SEM.