“Ideally, an OOO should include the basics,” says work and wellbeing psychologist Kate Sullivan. “The dates you'll be gone, an approximate timeline for when someone can expect to hear back from you upon your return, and how any urgent requests will be handled in your absence.”
I wouldn’t be offended or consider saying anything to anyone who included this in their out of office message, but even as someone who is in a religion that forbids use of electronics on most holidays, I still think this message is 1) TMI; 2) doesn’t convey what it needs to convey unless you are explaining that you don’t use electronics during holidays, in which case you can just include that you won’t be checking email without including the religious explanation; 3) would come across to me as inclusion of personal information I don’t need, which would therefore strike an unprofessional tone,and I wouldn’t understand why you felt the need to include that info. How about “I’m out of the office without access to internet or email until (date). If you need assistance before this date, please contact…” Like I said, I think your colleagues are overreacting, but in general I would advise to leave all personal information out of your auto-reply — vacation, medical leave, religious observance, etc. — people do not need to know why you are out.

My OOO is almost always “I’m out of the office and will be returning on x date.” My email sig has instructions along the lines of “for questions on x, email this list” for a couple of the more common areas people would contact me about that might need an urgent reply (as the lists go to more than just me, obviously), but most people have finally figured out to contact those lists to start with, anyway. If I were in the middle of a project or something that needed to be moved along in my absence, the project teams usually know when we’re out, but I’d put a back up in that case, if needed, but generally there’s not much to be gained by a longer OOO from me.
Our office VM sends an audio file to our email. I love it, because I can apply all the filing and prioritization functions of my email inbox to the voice messages. It’s the only way I ever check my VM. I don’t even remember how to listen to messages on my phone.