Drew - June is the most likely month for a packraft trip on Escalante. Good thing you are a constant planner, it will come in handy with this trip. Here is some basic information:
The gauge station is a good reference, but not much else. I would recommend you do NOT attempt a trip without the gauge station reading at least 5 cfs, I don't know where the general recommendation of 50cfs came from, but in a packraft that would be a fairly technical ride. The gauge doesn't show any of the major tributaries and their contribution to the flow is significant. At times Boulder Creek can double the river flow and Boulder is not measured and comes in at Mile 8 after the Highway. Pine creek is also a major contributor and it's confluence is after the gauge.
As for a put-in and take out location...you have so many options here it isn't even funny. Lots of packers enter Fence Canyon (just up stream from Neon you mention), but with good water you are cutting your trip in half. From the trailhead at Fence to the river is just over 3 miles, all down hill!
The most common take out is Coyote Gulch, using the Crack-in-the-wall as your exit. This is about 4-5 miles and is a climb. You could consider going the extra 10 miles to Lake Powell for a pick-up but will increase your costs significantly.
Depending on how much time you have there is an exit route off the river at no more then 10 mile increments. To do the whole river from Highway 12 to Coyote Gulch should be about a 7-10 day trip, depending on the time you spend floating. I know people that have completed the entire trip all the way to Lake Powell in 5 days - but honestly why?
Hit me up with any other questions you may have...I have lots of information and am working on publishing it at escalanteriver.blogspot.com. Or send them to me directly at email@example.com.
Good luck planning - JAC