I’ve had several co-workers over the years use “at my earliest convenience” and it’s always rubbed me the wrong way. It just sounds kind of dismissive like, “I don’t care when you need it, I’ll get to it when I get to it.”
Just say you’re away from your desk! Or if you’re out for more than a day, just say you’re “away” until XYZ date.

When you share transparent business information and provide alternative ways when the relevant channel is not available, it delivers a delightful service experience.
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.