The law says they can look into his personal email if it's on their phone. From a technical point of view, in order to see everything that's on his iPhone, they would need software to monitor it which Leo doesn't believe exists. They would have to ask for his iPhone, which they could do, and look at it that way, but Leo doesn't think they could spy on it remotely. They can do this on the desktop though, and even if he's using company Wi-Fi on his phone, they could.
If he's just using the cellular network and not company Wi-Fi, they cannot watch his traffic. Even if he were to send or receive email at home, once he got to work they wouldn't be able to see those emails. They would only be able to see the emails that go over their network.
If his email provider uses HTTPS, then the email would be encrypted and he would be able to examine the certificate. There is a trick, however, that businesses sometimes use -- a "man in the middle" attack where they capture all of the data using a fake certificate. But Barry would be able to tell they were doing that because on the desktop, the certificate would show up as the company's certificate. They could do that on the phone too, but he would know it. There is no Apple tool that would allow them to spy on his email. This would be a different scenario if it were a Blackberry or Android phone, however. Companies can spy on your emails if you're going through the Blackberry server, and the company could potentially install software to do this on Android.
Courts have ruled that if you are on a company phone making a call, the employer is permitted to pick up the party line and listen in until the employer detects something personal going on. He would not be able to evesdrop even though it's their phone and on their time. When it comes to digital conversations, however, they can look at anything. There is no right to privacy when it comes to digital technology. Legally, a company can spy on you without your knowledge and in fact, if you're working at a big company, chances are that they do spy on what you do.