In managed to get it working by using a http request node and by specifying the user name and the password in the URL (see screenshot above). For this I had to use the query string parameters "u=" and "p=" for the user name and the password.
Note that It didn't work when I selected "Use Basic Authentication" and specified the user name and the password there instead of in the URL. In that case I got following 401 HTTP response:
{"_msgid":"4587d1ac.fea2","topic":"","payload":"401 Access Denied - Failed to Authenticate (0)","statusCode":401,"headers":{"date":"Sun, 21 Oct 2018 12:08:35 GMT","server":"Apache","x-powered-by":"PHP/5.6.36","www-authenticate":"basic realm="DDNS"","cache-control":"max-age=2592000","expires":"Tue, 20 Nov 2018 12:08:35 GMT","vary":"Accept-Encoding","content-length":"46","connection":"close","content-type":"text/plain;charset=UTF-8","x-node-red-request-node":"eb2849ee"},"responseUrl":"https://nic.changeip.com/nic/update?hostname=boXXXXXX.onmypc.org&myip=87.67.108.163"}
