They want to because they had a referendum on it and it's legally binding to them, but the EU has told them that that means losing free market access (e.g., https://www.theguardian.com/world/2016/jul/03/eu-swiss-singl... ).
Apart from that tabloids will make a lot of noise around this, obviously it's great topic to get some news :-) In this uncertain time it is not that easy to make predictions, especially when it comes to such high level.
Both the UK and Switzerland have to allow people from the EU to travel into and work in their country, don't need to provide them with benefits if they don't have a job. In return UK and Swiss citizens have the same rights in the rest of the EU.
Both the UK and Switzerland can decide to allow people from outside the EU in whatever way they want.
In 2015 during the migrant crisis EU governments decided to relocate 160,000 refugees over Europe, but the UK opted out ( http://www.express.co.uk/news/world/607349/Migrant-crisis-ma... ) so this doesn't apply to them.
For me that causes a problem as I'm saving for a house in the continent, which I have to pay in EUR, now that I might have to leave the UK as my status is not guaranteed: https://hansard.parliament.uk/commons/2016-10-19/debates/F13...