Not that I’m considering it, but…
Is it just simply a matter of calling up your local parish priest and having him make a phone call for you? Do you get to choose your monastery and/or order (Benedictine, Franciscan, Jesuit, etc.)? Are you expected to have a certain degree of education?
Thanks.
Yes, you get to choose your order, and possibly your monostary. I think that you would actually call the monastery that you are interested in and submit an application.
How do they conduct the job interview if they’re not allowed to talk? 
Most monks are quite chatty. I don’t think there are any serious ammount of oath-of-silence monks left, but that’s just a WAG.
I’m sure the details of the process are different for the various orders, but in general, the following things might happen.
You could certainly talk to your priest, as likearock suggested, or if you have a good idea of which order interests you, you could approach them directly. Some orders focus on teaching, for example, others are contemplative. Some are cloistered.
After you’ve expressed your interest, most orders that I’m familiar with will encourage you to spend some time with them, to make sure it’s a good fit. This might start with weekend visits. As you move forward with the process, you would join the order formally, and begin a course of spiritual study that would result in the taking of vows. Some orders have various levels of vows, and strongly encourage you to speak up at any point in the process if you change your mind about joining. Obviously, they take the vows very seriously, but at the same time, they realize it is self-defeating to have someone stay if he has a change of heart about being a religious.
On the vow of silence part, most orders have certain times set aside for silence, for prayer and contemplation. From what I know, this varies from an hour a day (the monks who taught us in school) to almost all the time. Also, a monk can take it upon himself to undertake a period of silence. Some cloistered orders have a few monks (or nuns, if we’re talking about them too) who are designated to deal with the outside world … answering the phone, dealing with repairmen, etc, so that the majority of the community can remain in silence and/or contemplation.
One of my brothers was married by an abbot and he (the abbot) was quite the conversationalist. His abbey (I believe they are Benedictines) have one week out of the year reserved for total silence.
My brother and his wife were going to drop by for a visit (they live out of the area now), but they were told to come by after their week of silence was over because they would enjoy the visit more.
I imagine that nearly all of the orders have some sort of presence on the web now.
For example try http://www.orb.org
In most of the silent orders, the abbot or prior is allowed to speak, and may give other monks permission to speak. Since that’s probably the person you’d be talking to anyway, if interested in joining, there’d be no problem.
As to day-to-day life with the vow of silence, my mom was in a Trappist convent briefly in her early twenties (before she was married, of course), and she says the sisters had developed a very elaborate sign language for general chitchat.