I'm slave to a Windows laptop at work, and SQL Developer is a shitshow.
If you enter the wrong JDK, for example OpenJDK's J9 engine, then the piece of crap doesn't work.
There's a SQLDeveloper.conf inside the SQL Developer installation, with a SetJavaHome variable, but no, it's not that one. That'd be TOO easy. Instead, you got to locate which file it had last modified in your user folder, and that'll be in product.conf inside AppData\Roaming\sqldeveloper\{version}.
There, you can point ot to your Favorite JDK...
SetJavaHome C:\Java\OpenJDK8U-jdk_x64_windows_hotspot_8u181b13\jdk8u181-b13