If it's Google Apps for your Domain (GAFYD), your students would have an email account with the domain name. For instance, I have Freshmancomp.com as my domain. I signed up for GAFYD and gave every student an email account. This worked great with students who didn't already have an email account. Now they all already have email, so I just use regular Google and the accompanying apps. You have to add every student to your address book manually and they are then connected to you, but not to each other like GAFYD.
So to answer your question: No, students don't need "gmail" accounts, but they do need an email address.