I don't think there's such a thing as overqualification. And if you look at the job announcements for most Federal jobs, based on their outrageous requirements, Uncle Sam agrees with me.
Here's the thing, as I see it - the problem with much of EMS is not that your degree will keep you from getting a job, but rather that your degree will not be valued. I was just discussing this with a friend of mine the other day. EMS agencies tend to set your pay based on two major criteria: 1) You have a patch and a pulse; and 2) the year you gained your patch. Your additional education seems to enter into it very little.
Now, with a degree, you're well positioned to move up in the EMS world, if you are with the right employer. A small, private transport company might pay you a dollar or two more to be a "supervisor" - still driving a truck, but somehow higher ranked. On the other hand, a huge corporation like AMR, you would likely have more opportunity to move up into supervision, education, management, etc. with a degree than without. And, a degree gives you more credibility if you want to get involved in the larger EMS world, sitting on state and national committees, becoming one of the people who make policy, and so forth.
I'm not in a dissimilar situation. EMS isn't my career, though it's something I greatly enjoy. I have a Bachelor's degree and (arguably) a doctorate. I guarantee that if I worked as a basic, I'd be paid the same $12 an hour they pay around here. But, if I was willing to sell my soul to the local evil empire (BLSBoy knows who I mean! ;p), I could probably move up the chain faster than the medic with a GED.