With three games played in Pool Four, Warriors have no wins and just one bonus point on the board as they head to France to take on Castres on Sunday.
The return tie with the Top-14 side comes just nine days after Warriors delivered a season's worst performance in the 9-6 defeat at the hands of the Frenchmen.
With Glasgow skipper Al Kellock rested because of the five-game playing protocol, Hastings has questioned the comfort zone in which players in the PRO12 professional era are now operating.
The great Grand Slam-winning centre will be providing expert analysis for Sky Sports at the Stade Pierre Antoine on Sunday, and fears that another no-show by Gregor Townsend's troops will undo all the good work that has gone before by Glasgow in France.
With the English Premiership pushing for a reduced PRO12 representation in Europe's top-club competition, Scottish woes this season are doing nothing for the retention of Heineken Cup places for both Warriors and Edinburgh.
Hastings said: "For me the Glasgow players have to go to the Stade Pierre Antoine with the attitude that they are scalp-hunting.
"This is a massive game for Glasgow because, if it goes badly, then an awful lot of the good work Glasgow have done on behalf of themselves and Scottish rugby in recent seasons will be left in ruins.
"I am thinking of the great win at Toulouse three seasons or so back, the draw at Montpellier, and the home victory over Biarritz. But if Glasgow perform at the same levels they did in the first tie with Castres, which Gregor (Townsend) himself said were wholly unacceptable, and they take another sore one, then it could consign them to the red button margins when it comes to European TV games.
"At a time when the Scottish clubs' representation in the Heineken Cup is under fire, neither Glasgow nor Edinburgh have done anything to state their cases. All of which leaves one to ask if the comfort zone of the PRO12, where there is no relegation, is just making it too easy for the modern player.
"But that is by-the-by. If Glasgow can go to France and play with their hearts on their sleeves, and show real pride in the jersey, they can get a result that will regain their honour and set them up for the 1872 double header with Edinburgh.
"Whatever way you look at it, this is a huge game for Glasgow."
A huge game it may be, but it is one in which the leadership of Warriors skipper Al Kellock, who is rested after reaching the five-consecutive-game ceiling, will have no influence.
Hastings said: "I look down at the likes of Courtney Lawes, the Northampton lock, and if the Saints need him to play more than five games do you think he would be rested?
"Glasgow need Al Kellock on the field in France on Sunday and he will be sitting in the stand."
Hastings was also frank in his assessment of how important an away win would be to Warriors head coach Gregor Townsend, in charge of his first Heineken Cup campaign.
The Scotland legend said: "While the players must find a way to focus for the full 80 minutes, and not just, say, 36 like they did at Northampton, ultimate responsibility rests with Gregor.
"Every week is a big one for Gregor as Glasgow head coach, but this is probably his biggest yet."