Yes, public opinion supports deporting illegal immigrants, because the public, as a whole, wants law enforcement. The public supports legal immigration, and wants to make that easier, but they also want security, and want to see that laws are in place and enforced when they are violated. Yes, the public supports enforcement of deportation, because they want the nation to be secure.
Public opinion does not support the deportation of illegal immigrants as our country was mainly founded on illegal immigrants. Most of American just wants to work and get their jobs done, and does not care who is involved in that process. They understand the need to work and understand that it is hard to get where you need to go in life.