Talking about Syria as there is only one church is a misrepresentation of the reality. Indeed, in Syria there are around 13 Christian denominations. These can be generally divided into three main groups. The Orthodox ones (Greek, Syriac, Assyrian and Armenian), the Catholic ones (Melkite, Syriac, Maronite, Chaldean and Armenian) and Protestants (there is a couple of them).
Each of these churches has its own leadership. Indeed, the attitude of the Churches' leaders from the revolution is not uniform. While the Catholic churches supported the regime openly because of their fear of chaos, the orthodox churches showed somewhat explicit support to the demonstrators. I have no clear idea regarding the position of Protestant churches.
The Christian public attitude is still split. The extent of Christian public support to the revolution varies according the region. While it is relatively low in Aleppo and Damascus, this support reaches 80-90% in some areas, including Homs, Hama, Damascus suburbs and the west coast.
Finally, I would like to emphasize that in Christians' political views and attitudes in Syria there is complete separation between Church and State (from this point of view, there is clear difference between Egypt and Syria).