“Happy Holidays” is typically used when you aren't really sure what holiday someone celebrates. In that instance you are replacing Christmas or Hanakkuh with the word Holiday making it a proper noun, which means it should be capitalized.
The start of the holiday season does not mean letting your inbox enjoy the same degree of leisure and festivities as you do. Despite being busy with all the buzz and planning, you can hardly lock your business’ door before taking care of what happens with your inbox.

Yes, I phoned a dentist office late in the day for a reinfected root canal problem and got a cutsy “humphrey bogart” fake reply on their voicemail — I thought it was extremely inappropriate for a business office to use something like this. It was hard to find it funny, especially because while calling me “sweetheart” and all that, the message didn’t actually tell me when they might return my call (later that day? Next day? Next week? Never?) nor did they offer any options for emergency contact with another dentist.
Coworkers, clients, and subscribers typically expect fast responses and solutions to their problems, particularly from people working in customer service, marketing, and communications jobs. Out of office messages provide them with a polite, concise, and professional explanation of why you cannot respond right away. You can provide an OOO message if you are gone for one day, one week, or several months.