I am on sick leave with no access to emails and phone calls. Hence, kindly expect a delayed response.
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.

The more information you include in your outgoing message about the colleagues who can assist in your absence, the less likely you are to interfere with ongoing projects. This will help ensure you enjoy that well-earned vacation.
I will be out of the office starting (Starting Date) through (End Date) returning(Date of Return). If you need immediate assistance during my absence, please contact (Contacts Name) at (Contacts Email Address). Otherwise I will respond to your emails as soon as possible upon my return. Thank you for your message. How do you respond to holiday greetings? "And (also) to you!" You too! Thanks, same to you! Happy New Year to you and yours (when you want to extend the wishes to the other person's family) How do you let clients know you are going on vacation?