There isn’t currently an option for MegaReader to save the username and password. However, you can include the username and password in the URL when you add the catalog to the Book Catalogs screen. In this way, you will be automatically logged in when you tap on the catalog link.
To do this, enter the URL in the form of http://username:password@address. For example, to connect to a Calibre content server on my local network with a username and password, I would use the URL http://patrick:mypassword@Patricks-MacBook-Pro.local:8080/opds.