One problem with every individual following their own conscience is that consciences can be wrong. While we are told "let your conscience be your guide," almost all religious groups understand that this advice must be tempered. This is why religions appoint people with authority whom, traditionally, one is expected to consult before acting—or in more equalitarian groups, such as Quakers, one is expected to consult with the group as a whole before acting. This helps prevent people from rash actions that might do more harm than good. It also acknowledges that no one person is God: we all have blind spots and can benefit from consulting with others.

We are an individualistic society, and as I write the above, I am aware that transcendentalists like Thoreau and Emerson would vehemently oppose, at least in theory, relying on authority or other people in making a decision, arguing that this dilutes and corrupts the purity of the individual mind, heart, and conscience and paralyzes good action. This also contains truth, but a deeper truth is that individual action is usually far less effective that a coordinated and disciplined group plan, such as Dr. Martin Luther King Jr.'s civil rights movement or the gay marriage movement of the early twenty-first century. People going off doing what they want for the individualistic reason of taking a stand can do more harm than good to the causes they hope to support.

Ironically, King was thinking of Thoreau's notion of civil disobedience when he wrote his "Letter from Birmingham City Jail," but King had the wisdom to yoke the tradition of civil disobedience, which started long before Thoreau, to coordinated and carefully planned group actions.

The real problem comes, however, when a person whose "conscience" has gone awry is told by it to do things like shoot up a black church to start a race war. A society in which too many individuals simply start doing what they think is right is threatened with chaos.

By way of counterargument, one might consider what happens if a person who follows their conscience to break an unjust law has people who depend on them for their survival.  For example, if a mother of two children decides to break an unjust law, according to Thoreau's argument, she must be prepared to accept the consequences of that law-breaking and go to jail.  First, she will incur legal fees.  Second, any income that she provides to support her family will be lost.  Now, if she is the sole breadwinner in her family, or even a single parent, what will become of her family as a result of her breaking the law?  Does she have more of an obligation to break the law or to continue to support her family and provide security for them? 

Thoreau spoke from a relatively privileged position: he didn't have a family who was depending on him for their livelihood, and he did have friends on whose land he could live or with whom he could live for a time (Emerson, both).  He could refuse to pay his poll tax and go to jail, and no one is harmed by his behavior.  However, if he were responsible for others, he might feel differently.  It is possible for people to be torn between two impulses in this way, and Thoreau's argument fails to consider that some people have other responsibilities that are just as important to them.

Thoreau, himself, will not provide much in way of counter- arguments to his points in Civil Disobedience.  Instead, I think that you might have to consider the consequences of Thoreau's claims and extrapolate what might happen outside of Thoreau's work.  For example, on the most basic of levels, if we embrace the idea that if one disagrees with the laws then breaking them becomes a moral responsibility, one could literally shudder at such an idea.  It is not a major problem in the context of the work because Thoreau's moral compass is not threatening.  The reader understands that Thoreau's sensibilities embrace inclusive, participatory democracy and do not pose an immediate threat to anyone except those in the position of power, individuals that he has cast as someone who has lost their moral compass.  Yet, consider for a moment if someone who lacks Thoreau's moral compass follows his advice.  Consider those who kill doctors who perform abortions or those who bomb health clinics that are linked to abortions.  These individuals "follow their conscience."  Yet, I don't think that what these individuals do are in concert with "forming a more perfect union" or "securing the blessing of liberty" in the sense of the Constitution.  Everyone is threatened in a setting where individuals fore go the law by breaking it.  Thoreau operates under the perspective where individuals who have a conscience would share his own view of what that conscience might be like.  I think that this is a rather compelling argument in why his beliefs could be disastrous, if placed in the wrong hands.