github’s documentation seems to be rather coy about this, but I believe that the following statement is true: sometime in the past couple years, github changed their policy and began to mandate that users create SSH key pairs to connect to it, rather than suggest that they do. (Using SSH keys for this is absolutely best practice, so I’m largely in favor of the decision. I do wish that Googling their documentation came up with something clearly stating this, however.)
Bottom line is: each developer on your team must follow Ian’s excellent instructions to set up an SSH key pair and use it to connect to github. (Begin with step 2 under “Making a github user.”)