Any way to get JBidwatcher working with fedora core 6's version of Java?

Try downloading the Sun Java runtime, and pointing directly to it.

In other words, if it installs to:
/usr/java/jdk1.5.0_09

you'd run JBidwatcher with:
/usr/java/jdk1.5.0_09/bin/java -Xms256m -Xmx768m -jar JBidwatcher-1.0.jar

Make sure you're using Sun's stable Java, not libgcj, OpenJDK, or IcedTea.