I once worked for a psychopath who contractually required all employees to answer their company-issued phone at all hours. He apparently did lots of drugs and never slept, and would regularly call and berate people at 3am. He was way beyond the setting boundaries limit, so I left in three months. I'm still amazed at the people who tolerate that kind of abuse for years.
Your husband's boss is simply leaving emails, which is nowhere near that level. I think there are a few levels of issues here.
The first question is why he has company email on his phone to begin with. If it's his personal phone then the problem isn't really the boss but his own addiction to work email. I've been there. Try what I did. Permanently turn off work email on all personal devices. On the iPhone (and I suspect other devices) you can still receive meeting invites even with the email off so that's no excuse. If he needs to read an email at home, take the time to login using a computer web browser. If he can't handle that and feels too cut off, deal with that core problem before blaming the boss. Personally, taking this step a few years ago was the best thing I ever did to reclaim my sense of self outside of work. I'll never go back.
If it's a company issued phone, that's the second layer. When there are core hours and it's not an on-call situation, disabling email away from the office is a good option. The stronger solution is to simply leave the phone at the office. Force them to call him on a personal number if it's really that important, and force him to physically leave work at work. Again, if he can't handle that and feels too cut off, deal with that core problem before blaming the boss.
If it's a company phone and he's required to have it active and with him and immediately respond at all times even in a non-critical role, then we're finally to the abusive boss phase. At that point the issue of boundaries and potential new jobs comes up. He needs to push back. But if it's really just more about your husband's inability to mentally leave the office, then even changing jobs may not fix the problem.