I had a job that was like that. First day of vacation, I kept getting bugged. Second day, I turned off my work phone, found that HR gave people my home number because HR thought I needed to be more responsive while on vacation. I then my home phone off, and went to my "friends/family line."
When I came back in, I only marked off the days that I was not interrupted. The boss gave me guff about it. Two days later, I resigned.
The ironic thing is that I had the boss call my home number demanding, "These tickets don't do themselves, resigning is no excuse." There were a ton of un professional things I could have said... but I didn't, just in case he was trying to goad me to say something that he was recording. I just hung up, blocked him and anyone from that company, and moved on. I had people leaving voice mails over 6-12 months after leaving there still upset that their tickets were not fixed.
Simple rule: If they demand "pitching in", every day you have to interrupt your vacation, means a day of comp time. If they don't understand that they are pushing boundaries, then "pitch in" your three envelopes, and find a job that respects you.