Looks to me like she isn't so quiet supporting Hillary. But the Bush dynasty members are all in with the Clintons it seems and it stands to reason that Dana Perino, their friend and colleague, would be of that mindset too and her participation at the Clinton Global Initiative Conference pretty well confirms it.
Nothing wrong with it. The Constitution allows us all our opinions and beliefs re politics, religion, socioeconomics or anything else we think or believe and we have liberty to associate with whomever we want whether we are right or wrong.
But it does mean Perino doesn't hold much in the way of conservative beliefs. It would be interesting to see her opinion of Hillary's reeducation camps proposal but I'm guessing she won't comment on that at all.