Mascherano was left out of Liverpool's Premier League loss to Manchester City amid interest from Barcelona - Roy Hodgson, the Reds manager, described the Argentina midfielder as not being in the right frame of mind to take part at Eastlands.
Begovic then said he did not want to be considered for Stoke City's Carling Cup win over Shrewsbury Town after rejecting a big from Chelsea for him.
Taylor, who is an Iron fan but made his name in football away from North Lincolnshire and was manager of the England team, compared their actions to treason.
"Both players should - and must - be punished for their behaviour, and the players' union, the PFA, should support whatever action the clubs wish to take," he wrote in the Daily Express.
"The standing and reputation of professional football and footballers are constantly taking a public battering. A lot is undeserved, but effectively refusing to play for your club is tantamount to treason - and the punishment should fit the crime.
"In their minds it is all about their own interests. That they are contracted to their clubs to play for them does not enter their thinking. Their actions are a disgrace to their profession."
Peter Coates, the Stoke City chairman, said on Thursday morning that his club would reject any further approaches from Chelsea for the Bosnian player.
Barcelona were given until Sunday to up their offer for Mascherano to the £22million price tag Liverpool want to let him leave Anfield.