How to deal with student putting their (home)work on github

While using github for source code is generally something I love to encourage, if a student puts their (computer science) homework there, it's generally easy for others to find and copy - which creates a temptation to use it as a "baseline" for their own (identical in most cases) homework - while I understand the benefits of using github (versioning, transitioning across machines easily, teamwork-capabilities), and the individual student who is using it has verified that it is indeed their account and can explain the code well, I still feel uncomfortable with it. Unfortunately I also don't know of any better solutions for them, as the non-public github is paid, dropbox does NOT work that well for source code (i.e. comparing versions), not teamwork (changes entire file, not line by line).

Has anyone else dealt with this issue? how did you handle it? does the university have some sort of policy around publishing student-created work openly? (even if it is part of an assignment)?

As a side-note, my general policy regarding "very similar" assignments is that whomever submitted it first gets the points, and the other submissions do not receive any points, however I tend to ask the students to explain their code and how it works, why they chose X over Y, etc. in such cases first.

note: I have not yet asked the student to do anything other than verify that it is indeed their account, not expressed my concerns, as I would like to think about it first. It certainly won't be the first nor last time I receive similar-looking assignments, but I seem to be unaccustomed to the "openness" of it as-is.