I think this makes a lot of sense for a 2-3 month absence, when there wouldn’t be much point in reading and responding to things when you get back. Questions will have been answered and issues resolved by different means.
Have you ever considered how your office design could be having a negative impact on the way your employees work?

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.
Hahaha. This sounds like somebody thought the phrase “at X’s earliest convenience” sounded vaguely businessy and professional, but didn’t realize the pronoun is always supposed to be “your”. It’s never “my”, for the reason you mentioned.