I think you are misunderstanding the relation of Java SE (or "Core Java) and Java EE.
Core Java is important to learn and is what most schools will teach you.
Since programming revolves around using wheels rather than reinventing them, there are usually libraries with code for accomplishing complex tasks.
For examples, some schools teach people to build GUIs using the AWT or Swing class library that are supplied with Java, although there are other options.
Java EE, while supported by extra software to run the program on servers, can be thought of more as a library of classes and related services. You still do Java constantly, you just use a popular and Sun supported library. You will also see Hibernate and Spring that are not part of Java EE.
Most places will first of all want to make sure that your Java is solid. As you start working, you will gain experience at using specific parts of Java EE.
