Did the Swiss Air Force assist Austria in self-defence during the Yugoslav War?
I'm likely reading the same sources than you read. The chance of the event you are searching for ever occurred seems slim. As you wrote, Austria worked hard to maintain its neutrality policy.

From Wikipedia, it seems clear Austria was not using guided missiles before 1993, but indeed other weapons can be used to shoot down an enemy. 

The article seems to be exhaustive for significant elements related to flights over the country by foreign aircraft, and still doesn't mention any destruction between 1970 and 1993.

There was a major incident in 1991 with Yugoslav MiG-21, but no mention of a shot down. This incident and the similar ones the same year led to the adoption of guided Sidewinder missiles by the air force two years later.


  From 1970, Austria purchased a total of 40 Saab 105 lightweight
  multi-role aircraft with the intention to deploy them in trainer,
  reconnaissance, interception and ground attack roles. As it became
  clear in the 1980s that the sub-sonic aircraft were inadequate for air
  combat and airspace interdiction, Austria purchased 24 reconditioned J
  35 Draken fighter aircraft to supersede the Saab 105 as the Austrian
  Air Force's main interceptor in 1988. The Saab 105 remained in service
  as a trainer/surveillance aircraft.
  
  Shortly after, the Draken saw their first major use in airspace interdiction starting 1991 during the Yugoslav Wars, when Yugoslav
  MiG-21 fighters crossed the Austrian border without permission. In one
  incident on 28 June a MiG-21 penetrated as far as Graz, causing
  widespread demands for action. Following repeated border crossings
  by armed aircraft of the Yugoslav People's Army, changes were
  suggested to the standing orders for aircraft armament.
  
  Since 1955, Austria's armed forces had been forbidden to operate any
  guided missile system, including Air-to-air missiles and
  surface-to-air missiles. In the post-Cold War environment, and with
  gun-armed aircraft a relic of a past age, the Austrian Parliament
  voted to amend this section of its state treaty and in January 1993
  modern AIM-9 Sidewinder missiles were ordered from Sweden to arm its
  fighter aircraft.


On the other assumption that Switzerland provided military assistance, again no mention of such cooperation until recent year, with the EU treaties.

It seems the overflights of Austria by non authorized aircrafts was quite intense --daily-- at the time of the Yugoslavia war.

You may be right when saying that Austria could be reluctant to disclose a shooting. The fact that it would have happened before the widespread use of Internet is not something that could prevent the event to be recounted online today.

Not relevant downing:


Jan. 1992: Two EU helicopters downed by Yugoslav MiG-21, Austria radars detected the downing.


Additional documentation:


The Austrian Air Force - In the Past and Today
Austrian Air Force
Austrian Air Force
A well documented list of interceptions over Austria
Austrian Draken's on high alert