In most cases, you probably want your code to be owned by root, but to be world-readable, and for scripts be world-executable. The Dockerfile COPY directive will copy in a file with its existing permissions from the host system (hidden in the list of bullet points at the end is a note that a file "is copied individually along with its metadata"). So the easiest way to approach this is to make sure the script has the right permissions on the host system:
# mode 0755 is readable and executable by everyone but only writable by owner
chmod 0755 healthz.sh
git commit -am 'make healthz script executable'
Then you can just COPY it in, without any special setup.
# Do not RUN chown or chmod; just
WORKDIR /app
COPY ./scripts/healthz.sh .
# Then when launching the container, specify
USER node
CMD ["./startup-production.sh"]
You should be able to verify this locally by running your container and manually invoking the health-check script
docker run -d --name app the-image
# possibly with a `docker exec -u` option to specify a different user
docker exec app /app/healthz.sh && echo OK
The important thing to check is that the file is world-executable. You can also double-check this by looking at the built container
docker run --rm the-image ls -l /app/healthz.sh
That should print out one line, starting with a permission string -rwxr-xr-x; the last three r-x are the important part. If you can't get the permissions right another way, you can also fix them up in your image build
COPY ./scripts/healthz.sh .
# If you can't make the permissions on the original file right:
RUN chmod 0755 *.sh