Mandatory vaccination

They're not agreeing on what the issue is. The argument made by John is one that ranks liberty higher than the common good. So the exchange would look as follows:

John: Mandatory vaccinations takes away the right to decide for oneself. Liberty is very important

Michael: But we took away the right to decide to take substances and this lead to societal benefits. Hence, the common good is more important.

Avoiding the Issue 

Exception

Michael could improve his argument by saying that John should factor in the common good. 

I see through my rose-colored glasses, that sometimes the cautions are relevant and sometimes they're not. If you can't tell, it comes down to who/what is believed, don't you think? If I'm punctured by a rusty nail from a hog farm fence, I'll examine my tetanus vaccination record.

''=

Acting in good faith, I would not say there is anything wrong with this exchange. John appears to value personal liberty above the common good, and Michael values the common good over personal liberty.

Being more critical...

John appears to be creating a rule that states that which violates [and] infringes on our liberties should not be allowed. Michael uses a form of the Reductio Ad Consequentia noting that we don't follow this rule for the common good. The problem is, it is likely that John doesn't agree with the fact that we don't follow this rule. For example, we don't make heroine legal. I would bet that John would think that we should, as he made clear personal liberty is paramount. If I were Micheal, I would argue the "Your Liberty To Swing Your Fist Ends Just Where My Nose Begins" perspective, pointing out that his "liberties" would kill other people (specifically those who cannot get vaccinated).