Email addresses have been available on the website, at every place I've been employed. I don't see why a work email address should be any more private than a work phone number- can you explain why you feel they should be private?
From the original email, it looks like this employee works off of trouble tickets that most likely come in through some form of service management software. They are most likely required to handle incidents only through this system, that way they can effectively be measured as to their productivity, as well as build up a knowledge base for issues that have a recurring theme or problem and the solutions for these problems. One technician may spend a week trying to fix a problem, once the problem is fixed the solution is documented, the next technician won't have to spend a week on the same problem. (that's the theory anyway).
By giving out his personal email rather than having submitted a ticket into the queue, it circumvents this process. If the technician were to handle problems only on a personal email basis, then no one would ever really know what anyone was working on. A highly productive employee could be looked at as a slacker if there is no documentation to show what they have been doing.
So if the person wanted their personal email so they could contact them directly to help with a problem, they definitely should not be given the email address. I would also call this person a SS for trying to work around the system.
modified to add the first paragraph