I was using the incorrect SAP jar. I was using sapdbc.jar when I needed the HANA client jar (ngdbc.jar). It all connected after that jar and driver switch.
try {
Class.forName("com.sap.db.jdbc.Driver");
String url = "jdbc:sap://xx.x.x.xxx:30015/?databaseName=DBNAME";
String user = "user";
String password = "password";
Connection cn = java.sql.DriverManager.getConnection(url, user, password);
ResultSet rs = cn.createStatement().executeQuery("CALL MY_SCHEMA.STORED_PROC");
// ... do whatever with the results ...
} catch (Exception e) {
e.printStackTrace();
}