I have listed mostly of my contacts with with the birthday dates and I con regularly view it in my google-calendar widget, but I have no idea on how to get a snooze on that day.
I use a fantastic Nexus4 (LG e 960) with android 4.4.2 (kitkat)
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While I don't know any built-in solution, there are several birthday reminder apps available on the Playstore promising exactly that, e.g.:
The 3 example apps I've picked above use your Google contacts to check for birthdays, so you don't need to manage a separate list. Others might use their own databases, so check their descriptions. There might be "privacy issues" when those apps have access to your contacts (which is needed for the birthday reminders) and full internet (e.g. for ads) at the same time, which you might wish to keep an eye on while picking a suiting app.