I have an RNA-seq dataset with 3 treatment groups, where gene g has group means mu_{g, i} for i = 1, 2, 3. Is there a way to use baySeq to calculate the posterior probabilities that mu_{g, 2} > max(mu_{g, 1}, mu_{g, 3}) and the posterior probabilities that  mu_{g, 2} < min(mu_{g, 1}, mu_{g, 3})? Specifications like groups <- list(null = c(1,1,1,1,1,1), alternative = c(1,1,2,2,1,1)) and groups <- list(null = c(1,1,1,1,1,1), alternative = c(1,1,2,2,3,3)) do not answer this question with the package's usual functionality demonstrated in the manual and the vignette. If not with baySeq, which packages can accomplish this easily?

No, you cannot compute posterior probabilities for this hypothesis using baySeq or any other package.

People typically take a simpler approach to this sort of question. One simply conducts pairwise tests between the three groups, and then selects genes that are (i) up in group 1 relative to both groups and 3 or (ii) down in group 1 relative to both groups 2 and 3. This sort of analysis is done every day using any of the RNA-seq analysis packages such as edgeR, limma, DESeq, baySeq.