Clojure and JDK11 allow you to catch signals, through sun.misc.Signal, (despite years of threatening to take it away).
So you can catch SIGHUP with the following:
"Catch HUP, INT, or USR1, etc. and run a one argument handler that
accepts a sun.misc.Signal"
(proxy [sun.misc.SignalHandler] 
If you need to send a signal from Clojure, I'd suggest shelling out, as to the best of my knowledge, the JVM only lets you send SIGTERM and SIGKILL:
(clojure.java.shell/sh "/bin/kill" "-s" "SIGINT" "<some-pid>")
(clojure.java.shell/sh "/usr/bin/pkill" "-HUP" "some-process-name")
Here is the relevant Kubernetes information with the pod configuration, which basically amounts to setting shareProcessNamespace: true at the top of the pod spec, and adding the following to the container that will be sending the signal:
securityContext: capabilities: add: - SYS_PTRACE
Also, if you've come this far, you should take a look at https://github.com/pyr/signal
Note that java has a flag that seems to be related. I found it after I wrote the original post.
Enables installation of signal handlers by the application. By default, this option is disabled and the application isn’t allowed to install signal handlers.