“Sad sad sad day,” Brown said in a post on his personal Instagram.
“We all had faith in Gordon and believed in making the Euros.
“14 points out of 18 in 2017 momentum was on our side.”
Strachan succeeded Craig Levein in the role, initially to oversee the end of the 2014 World Cup qualifying campaign.
Scotland’s attempt to qualify for Euro 2016 began brightly but they ultimately fell short of reaching the finals in France.
Strachan and the SFA board agreed he would stay on to lead Scotland in the 2018 World Cup qualifiers, but a 2-2 draw in Slovenia on Sunday ended their hopes of finishing second in Group F and reaching the play-offs.
The Tartan Army’s next campaign kicks off in 12 months when the inaugral UEFA Nations League gets underway.
Davie Moyes is the early favourite to take over at Hampden with Paul Lambert, Alex McLeish and Malky Mackay all among the front runners to take over from the former Celtic and Middlesbrough boss.