Neuquen is a city in Argentina in the province of Neuquen, where hitchhiking is generally hard. Neuquen is considered the northernmost province of Patagonia.
East toward Bahia Blanca
There is a peaje in Cipoletti 5 kilometres from Neuquen, but you will likely be sent away by the people working there, and don't expect to get a ride.
(January 2015) a good option is taking a bus from Neuquen Bus Station (also stops in Cipoletti bus station) to Villa Regina(is the last stop). You buy the ticket in the station and cost $35 pesos. I dont remember the number but ask there, is a 90km ride and takes at least 2 hour. In Villa Regina you can hitchhike at the end of the city, after the last light, once you are already on the road. Is not the best spot but cars can stop there, and if you can get to Chinchales(13km) you have a perfect spot, police control so the cars are really slow and plenty of space to stop everything,
There's an YPF at the edge of town where you can ask drivers, but traffic passes pretty quickly, so thumbing it probably won't get you far. If you can get a ride to Plottier, a town about 10km down the road, there's a stop light right before the highway begins again. If that doesn't work and you can get a short ride, you can try the YPF petrol station in Arroyito. Otherwise, from the city center in Neuquen (Sarmiento street) you can take the Koko bus that goes to Senillosa (maybe $25), and you get off when you get to Senillosa as soon as it crosses the main road next to the traffic light. From there it shouldnt take you long to get a lift.
You can easily sleep in the Bus Station without problem, expect cold.