We've identified a couple of bugs based on your message. They are:
1) The resolution of public host names test always fails if run, so it produces a false error message. Normally if the previous test to connect to VMware.com succeeds, this test won't be run, but if it does run, it will always fail even if you can reach the cloud storage provider.
2) The message "Unfortunately, Support Assistant will not work at all without client connectivity to VMware.com." is erroneous as well. It always shows up if any diagnostic test fails, even if your client connectivity is fine.
Unfortunately, these bugs do not explain why you are having an issue connecting to vmware.com. To figure that out, please file a Support Request via my.vmware.com.
Thanks!