what is Java (TM) Update Checker? Zone Alarm tells me that it is
trying to access the web.

Hello Affa4321,

The Java update checker is an application that can be set up to
automatically update your version of the Java run time system. See
  http://www.java.com/en/download/help/5000020700.xml
for documentation about the 1.5 version and
  http://www.java.com/en/download/help/java_update.xml
for the older (1.4 version). The message you are seeing from Zone
Alarm is indicating that the update manager is trying to access Sun's
java site for updates.

For a description from another use who also uses Zone Alarm is at
  http://www.javatester.org/installing.html
which describes the steps / promts he went through in installing Java
on his system. After reading this carefully, it appears some of his
tests had the updater disabled & others enabled.

There is also an explanation at "Audit My PC"
  http://www.auditmypc.com/process/jucheck.asp
indicating the Java Update checker is not spyware.

To find more references, try searches using
  java update checker
  java update checker spyware
  java update checker zone alarm
and similar phrases

If you find the answer unclear or incomplete, please make a
clarification request - I would be glad to answer more fully.

Good luck with your work.
  --Maniac