Can a middle-aged person get chicken pox?

Yes: Most adults have natural immunity, as the vaccine against chicken pox was only licensed in1995. But a small number of people even in the pre-vaccine era did not develop immunity by acquiring infection, so you could still get chicken pox if you were exposed and were not naturally protected by your own immune system. Little worry- most children are immunized so chicken pox is much less catchable now.

Yes & it's bad: Adults who somehow avoided chickenpox in childhood and haven't had the vaccine are at risk. Adults can develop a fatal chicken pox pneumonia as part of their primary illness. Blood tests can reveal whether you had it and are immune & if not i'd get the vaccine.