Jon,
Sorry that you are facing issues in MyEclipse 2015.
With MyEclipse 2015, the server connectors are updated to the WTP Framework. You can now create a new server from File > New > Other > Other > Server, which will also create a runtime under Window > Preferences > MyEclipse > Servers > Runtime Environment.
Can you please clarify whether the runtime environments were created in MyEclipse 2013 or MyEclipse 2015 ?
Can you try switching to a clean new workspace, create a new server (as mentioned above) and finally import all your existing project into the new workspace and try deploying the application in to newly created server.
If you are still facing issues, can you please attach the screenshots of the issue to help us get a visual of the issue. If there are any errors of interest logged in <workspace dir>/.metadata/.log file, then please attach the .log to help us investigate further.
Let us know how it works for you.