Sure Bob. The counter was the new JR model, but that actually turned out to be irrelevant. I started having the situation where the counter itself started rotating around the axle until it would go all the way around and bump into the fork. At this point it was facing down.
It seemed to be binding the wheel so that the wheel would not spin freely. After much fiddling I finally just took the counter off(not the first thing you think of doing when you're in the middle of a measurement) and discovered that the wheel still didn't spin freely. It turns out the axle was bad. I muddled through that day, and when I got home I bought a new wheel, and everything works great. My troubles had nothing to do with the counter.
But I'm still interested in my original question. If you do have a counter failure in the middle of a measurement, is it okay to replace it without recalibrating? Pete is exactly right that a new counter shouldn't have any effect on the calibration. But I think we all have the mindset that once you calibrate you don't want to change anything. I'm nervous even about taking my front wheel off and putting it back on. So even though it makes sense that changing a counter is not going to effect the calibration, it's something I would want to test out before actually doing it in the field in the middle of a measurement.