I ran the wilderness section three weeks ago in a 10.5 foot mini-max raft. When we put in it was probably around 350-400 and the levels fell significantly by day three (<300 according to AW). It was borderline boat abuse the last day. I would say the bare bare minimum in a boat that size would be 300, and keep in mind the flows on that river drop quick (i.e. don't put in below 400 if you want to boat out). Problem is the Gila gauge doesn't update consistently. We were R2ing, and I am 150 lbs and my girlfriend is only 100 lbs. If I had gone with a larger person and more beer we would have been dragging a lot. I had to get out and drag/push the raft maybe 5-10 times, mostly from operator error or choosing the wrong channel. It was still a blast, a beautiful wilderness run but at low flows with the headwind it was very slow going. It would have been doable in two very long grueling days in that boat so plan on 3. We had a hardshell kayaker and two packrafts and they seemed to have no trouble. A ducky would be the perfect craft probably down to 200. There are sweepers and strainers galore, but all very obvious and easily avoidable at those levels. At higher levels in a raft the current would probably be pushy right into the low hanging dead trees in a raft. They are not dangerous if you have eyes, just a nuisance. I would second stuntmansteve's comment that I would not take a 14 footer down it at less than 1000, and next time I do it will probably be in a ducky unless it is holding at above 500.
The Wilderness Lodge and Hot springs (gilahot.com) offers a shuttle service. It is a bed and breakfast and they didn't get back to us until a day after we took out because they put on the day before us. Evan of AW told me it was 175. Go to the Gila Hotsprings campground. It is a private campground with some of the nicest landscaped hot springs on the side of the river and it is cheap. It is a mile upstream of the confluence and you can put in there if flows are high enough or are in a hardshell.
I have asked on forums many times what the minimum flows are for this river are and 300 is doable in a 10 footer, but the bare minimum. Follow the weather forecasts to catch beatable flows. Two days of rain in the region will spike flows for a couple days so the window is very short.
This should be on every boaters bucket list.
"Ski bums need something to do in the summer, and river rats need something to do in the winter."