It looks like you just need to use the preservesinglequotes() function as CF will esacpe single quotes in dynamic sql statements. If you're executing your query like this,
<cfquery name="updateq" ....>
#sql#
</cfquery>
you would just change it to this,
<cfquery name="updateq" ....>
#preservesinglequotes(sql)#
</cfquery>
The only problem is that if any of your variables contain single quotes, you will have to manually escape them with the replace function like this.
<CFSET SQL = "UPDATE MasterOut SET MethodBy =""#replace(MethodBy_Value,"'","''","all"

#"", ....>
Is there any reason why you're trying to build a dynamic sql statement instead of just doing the query with dynamic variables? If you can get by without having to put the whole query in a variable, it will make this a lot easier.
Let me know if you still have problems.
GJ