Chris, here is the first part of your code:
<%check_var = CBool(rsEdit("Any_answer")) %>
<input type="checkbox" name="reply" value="true"<%If check_var = TRUE then response.write " checked"%>>
I.e. value of checkbox works if checkbox was checked. Then, insert below following code in second file (the form is calling) before SQL update statement:
<%
Dim sReply
If Request("reply")="true" Then
sReply=1
Else
sReply=0
End If
%>
And insert this "sReply" as value for Any_answer field, so in case checkbox is checked and it's value = "true", then sReply=1. Else, sReply will be eq. to 0.
Good luck!