You might want to check you datasource in your query.
<cfquery name="dbquery" datasource="register">
SELECT username,password FROM register WHERE username='#FORM.textfield#' AND password='#FORM.textfield2#'
</cfquery>
Where you have "register" as your datasource you also have register as the table you are pulling the username and password fields from. I ran the same code on my server with a little modification make sure your table fields are correct also. Here it is:
<cfif IsDefined("FORM.textfield"

>
<cfset MM_redirectLoginSuccess="clientsearch.cfm">
<cfset MM_redirectLoginFailed="no.cfm">
<cfquery name="dbquery" datasource="IGNEW">
SELECT user,password FROM access WHERE user='#FORM.textfield#' AND password='#FORM.textfield2#'
</cfquery>
<cfif dbquery.RecordCount neq 0 >
<cftry>
<cflock scope="Session" timeout="30" type="Exclusive">
<cfset Session.MM_Username=FORM.textfield>
<cfset Session.MM_UserAuthorization="">
</cflock>
<cfif IsDefined("URL.accessdenied"

AND false>
<cfset MM_redirectLoginSuccess=URL.accessdenied>
</cfif>
<cflocation url="#MM_redirectLoginSuccess#" addtoken="no">
<cfcatch type="Lock"><!--- code for handling timeout of cflock --->
</cfcatch>
</cftry>
</cfif>
<cflocation url="#MM_redirectLoginFailed#" addtoken="no">
<cfelse>
<cfset MM_LoginAction=CGI.SCRIPT_NAME>
<cfif CGI.QUERY_STRING NEQ "">
<cfset MM_LoginAction=MM_LoginAction & "?" & XMLFormat(CGI.QUERY_STRING)>
</cfif>
</cfif>
<title>index2</title>
<p>Please enter your login information below:</p>
<form action="<cfoutput>#MM_loginAction#</cfoutput>" method="POST" name="loginform" id="loginform">
<table width="300" height="160" border="1">
<tr>
<th width="155" scope="row">Username:</th>
<td width="155"><input type="text" name="textfield"></td>
</tr>
<tr>
<th scope="row">Password:</th>
<td><input type="text" name="textfield2"></td>
</tr>
<tr>
<th scope="row"> </th>
<td><input type="submit" name="Submit" value="Submit"></td>
</tr>
</table>
<p> </p>
<p> </p>
<p> </p>
</form>
<p> </p>