Hi, 

This is a common rust fungus that juniper get. It looks to be the Cedar-Quince Rust which is the type of rust that unfortunately can choke the branches and trunk of the plant and it can suffer from die back. There isn't anything you can effectively spray to control it since it is a naturally occurring airborne fungus. As you stated that it showed up right after it rained, that is exactly what happens. The spores float around in the air and once the conditions are right, with moisture and warmth, the fungus finds its host (in this case your Juniper) and gets to work. 

For discarding the plant, if it doesn't have value to you, you may want to do that before it dies, however you  won't really be able to eliminate the spread. If you have trees in the rosaceae family (serviceberry, hawthorn, apple etc) you don't really want to have any juniper around for cross contamination of the fungal rust diseases, but if you have a neighbor close by with a juniper, it won't matter anyways. 

Sorry for the disheartening news. 

Emily