Delete Loop with ListGetAt

Each checkbox has its own name as you can see in the attached code. it produces names as the following:
ProductGroupID1
ProductGroupID2
ProductGroupID3
so the name and the value is not the problem anymore? is it something on my action page?

No, uncorrected old cold still appears here and there. For example, in the line
<cfif Evaluate(Variables.ProductGroupID) NEQ "">

I would replace the lines

<cfset Variables.ProductGroupStateID = "Form.ProductGroupID" & #delProductGroupState#>
<cfif Evaluate(Variables.ProductGroupID) NEQ "">
WHERE Distributor_ID = #Evaluate(Variables.Distributor_ID)#
AND Product_Category = '#Evaluate(Variables.Product_Category)#'
AND Username = '#Evaluate(Variables.Username)#'
AND Authorized_State = '#Evaluate(Variables.Authorized_State)#'

respectively, with the lines

<cfset Variables.ProductGroupID = #Evaluate(Form.ProductGroupID & delProductGroupState)#>
<cfif Variables.ProductGroupID NEQ "">
WHERE Distributor_ID = #Variables.Distributor_ID#
AND Product_Category = '#Variables.Product_Category#'
AND Username = '#Variables.Username#'
AND Authorized_State = '#Variables.Authorized_State#'

I don't understand why it only contains one element, it should find all four right?
Wrong. Do the test I gave you, and see for yourself. If four checkboxes share the same name x, there will just be one value of form.x.

It is apparently unnecessary to end the value of ProductGroupStateID with & "|"

Check 2 or 3 boxes in the following snippet, and you should see what the matter is with your code.

<cfif isdefined("form.sbmt")>
<cfdump var="#form#">
</cfif>
<cfoutput><form action="#cgi.script_name#" method="post"></cfoutput>
<p>
<input type="checkbox" name="ProductGroupID" value="box1">chkbx1<br>
<input type="checkbox" name="ProductGroupID" value="box2">chkbx2<br>
<input type="checkbox" name="ProductGroupID" value="box3">chkbx3<br>
</p>
<input type="submit" value="submit" name="sbmt">
</form>

The checkboxes share the same name. Hence there will only be one value of form.ProductGroupID, irrespective of the number of boxes that you check. If you check more than one box, form.ProductGroupID will store the respective values as a comma-delimited list.

I would suggest that you differentiate the names of the boxes dynamically, as follows