Please use CustomField1, CustomField2... and so on in the xml tags instead of values like "Location" in other word your entry should be....
$entry = "<entry xmlns='http://www.w3.org/2005/Atom'>";
$entry .= "<title type='text'>". htmlspecialchars("Hmmm this is ignored right now") ."</title>";
$entry .= "<updated>". htmlspecialchars("2008-06-23T14:21:06.407Z") ."</updated>";
$entry .= "<author><name>". htmlspecialchars("Acme Industries") ."</name></author>";
$entry .= "<id>urn:uuid:E8553C09F4xcvxCCC53F481214230867087</id>";
$entry .= "<summary type='text'>". htmlspecialchars("Customer document") ."</summary>";
$entry .= "<content type='application/vnd.ctct+xml'>";
$entry .= " <Contact xmlns='http://ws.constantcontact.com/ns/1.0/'>";
$entry .= " <EmailAddress>". htmlspecialchars("test951@exaweff.com") ."</EmailAddress>";
$entry .= " <FirstName>". htmlspecialchars("First") ."</FirstName>";
$entry .= " <LastName>". htmlspecialchars("Last") ."</LastName>";
$entry .= " <CustomField1>". htmlspecialchars("Murrieta") ."</CustomField1>"; //<<This is the custom field that I want added and is not working
$entry .= " <OptInSource>". htmlspecialchars("ACTION_BY_CONTACT") ."</OptInSource>";
$entry .= " <ContactLists>";
$entry .= " <ContactList id='http://api.joesflowers.constantcontact.com/ws/customers/joesflowers/lists/2' />";
$entry .= " </ContactLists>";
$entry .= " </Contact>";
$entry .= "</content>";
$entry .= "</entry>";
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