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.
I will be out of the office starting (Starting Date) through (End Date) returning(Date of Return).

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.
STANDARD VOICEMAIL MESSAGE FOR ALL OTHER CMSD PHONES: • Every CMSD desk telephone and cell phone must be equipped with a voicemail greeting that is professional and concise and that conveys relevant and useful information to the caller. • All desk/cell phone voice mail greetings should include: o Name of employee. o Title of employee.