This is a great question. There are many ways in which to look at the connection between transcendentalism and Facebook. Let me give you some points.
First, transcendentalism espouses two main views. First, people are inherently good and organizations and structures corrupt people. Second, people and society are at their/its best when there is self-reliance. In light of these two points it can be argued that Facebook does and does not fit in.
On the one hand, it fits nicely with transcendentalism, because it gives a public forum for the exchange of ideas without constraints. There is freedom from institutions. In other words, people have their own voices. Some revolutions have been abetted by social media. This is an important point to keep in mind.
On the other hand, Facebook also shows the intolerance and evil in people's hearts. This does not show the inherent goodness of people. Recently there has been news about bullying on Facebook and social media. This would not fit in well.
In the end, it is safe to say that opinions will be polarized and the outcome will therefore be interesting.
There are a few difficulties with how this question is formulated.
The first is the problem of analogy. A good analogy is a comparison of things which are similar in the respect in which the comparison has been made. To compare oranges and grapefruit or oranges and apples as meal items or even oranges and red peppers with respect to Vitamin C makes sense, but if one were to compare oranges and scissors, the aspect to be compared would be unclear, and thus the analogy bad.
Since transcendentalism is a philosophy and Facebook an internet site, they are very different orders of entity and to compare them as did the poster above is to make somewhat of a category mistake.
One can ask how someone following the philosophical system of transcendentalism might regard Facebook. Since Facebook is a technology encouraging short soundbites rather than profound solitary reflection, and judging things based on masses of "likes" rather than detailed knowledge or introspection, a transcendentalist would probably consider its effects harmful to its users' development as thinkers and members of society.