Why am I getting an error when comparing trigger.oldMap with trigger.newMap on a boolean custom field?

Trigger.old is a Map of Case Id and Case instance.

To retrieve the field values, you will need to specify the ID as index and then the field name to get the corresponding values.

Trigger.oldMap.get(Case_ID).FieldName
Trigger.oldMap - A map of IDs to the old versions of the sObject records.
Trigger.newMap - A map of IDs to the new versions of the sObject records.
Trigger.old - A list of the old versions of the sObject records.
Trigger.new - A list of the new versions of the sObject records.
You may use Trigger.new, which is a list of Cases
for(Case newCase: Trigger.new){  //iterate through the list
    //Get field values you need
    System.debug('** Picklist : Case Status : ' + newCase.Status);
    System.debug('** Boolean : Case Closed by User : ' + newCase.Case_Closed_by_User__c);
    System.debug('** Text : Case Queue Name : ' + newCase.Case_Queue_Name__c);
    //For the comparison, do:
    //get old Case
    Case oldCase = Trigger.OldMap.get(newCase.Id);
   //Compare old and new value
   if(oldCase.Case_Closed_by_User__c != newCase.Case_Closed_by_User__c){
       //your logic here
   }

}