The documentation describing JBoss AS 7 Remote EJB Invocations mentions the fact that in JBoss AS 7.1 the security realm on remote EJB calls is enabled by default. Unfortunately, this means the example in the documentation doesn't work out of the box with JBoss AS 7.1 and EAP 6. The quick and easy way to resolve this is to disable the security realm in standalone.xml (which is mentioned, but not explicitly described).
Here is the relevant section as it occurs in the JBoss AS 7.1.1.Final distribution:
<connector name="remoting-connector" socket-binding="remoting" security-realm="ApplicationRealm"/>
Simply remove the security-realm="ApplicationRealm" attribute and you should be able to connect without a username and password.