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.
While injecting a little humor into your vacation email message can sometimes be a good idea – depending on your company and contacts – avoid oversharing and keep it professional at all times.

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Here’s my pet peeve: OOOs that specifically state the person “won’t have access to email.” It contributes to this pervasive idea that an employee who might technically be ABLE to check her work email while OOO better have a damn good reason why she won’t be doing so. Which calls back to the reason someone’s OOO is no one else’s business. Whether you’re OOO because you’re on your honeymoon, having your gall bladder removed, or robbing a bank, OOO should automatically imply unavailability for work stuff. Full stop.