Much of the reasons people cite for how illegal immigrants "destroy" America are myths. I happen to completely disagree with the suggestion that they are "destroying" anything.
The media and certain special interest groups have put ideas into people's heads, and it has worked. Unfortunately, special interest groups that are on the side of immigrants are ineffective at communication. This ineffectiveness is the result of failure to make the right argument with facts and evidence available. Some would suggest they are ineffective because they are wrong, but this is not the case. In some cases, their arguments hurt them more than they help. But this does not mean their cause is wrong.
Anyone who says anything absolute one way or the other is unwise to do so, because immigration and its effect on a recipient country the size of the United States is highly complex. I happen to believe there is more powerful arguments on the side of illegal immigrants, however this does not mean we should not enforce our immigration laws at the border and prevent further illegal entry.