<input type='checkbox' name='ckbx1' onClick='saveStatus("ckbx1");'>ABC

I then execute the following to set the check box to whatever it was
the last time it was changed:

<script type='text/javascript'>
document.getElementById('ckbx1').checked = getCookie('ckbx1');
</script>

This seems to work ok in IE6 but in Firefox a "...has no properties"
error is generated when the script executes. The same error is
generated when I click on the check box.

What is wrong is simply that you use getElementById(), but didn't assign an
id to the checkbox.
You only named it. Names are for formelements (name/value-pairs) so the
receiving script can use them.
ids are for JS.

Solution: add the id to the box.

The reason IE does it 'right' is because IE does it wrong, and confuses name
with id.