I work with such a person. So instead of something like “I’m writing to ask if you would graciously consider serving on our committee”, she will send a message along the lines of: “I am graciously writing to ask you to serve on our committee.”
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.

Setting up an out of office email depends on each company and the emailing systems and inbox setting used. But, it's also possible to set up an automatic reply for Gmail accounts and, with the coronavirus pandemic seeing more people go freelance than ever before, we're going to outline exactly how to get that Gmail out of office setting up and running.
That said, I do realize that I can’t just not use my phone at work. I’m surprised she has not been called on it by her coworkers yet!