There’s no law against it, but some employees may feel that announcing their birthday on public social media violates their privacy. While it’s great that you want to recognize your employees and celebrate with them, I recommend not announcing an employee’s birthday without first getting their permission.
In the case of announcements on public social media, I would get a signed acknowledgement that the employee has given you permission to share their birthday and that their doing so is completely voluntary. Announcing birthdays on public social media pages is a little riskier because they can be seen by everyone.
You could instead make the announcements internally (still with permission). Popular approaches include email, newsletter, or intranet. Many offices have a birthday celebration each month, such as a gathering with cake, ice cream, or cupcakes, and announce the employees who have birthdays during that month without mention of their exact birthday. This is a way to celebrate employees and increase comradely and morale, while avoiding shining too bright of a spotlight on any given employee on a particular day.