When you test the access by entering the UNC path to the SharePoint file in Windows Explorer, is that from your personal local computer or from the server where SSIS is running? The reason I ask is because I've had an identical situation with an SSIS package I've developed going to SharePoint 2010 using SQL Server 2008 R2 and processing files after first checking to see whether or not they exist.
Is the SSIS package automatically applying the credentials associated with your account or with a static built-in account? If the later is the case, then perhaps it lacks the required credentials. You may need to follow up with the SharePoint admin for assistance.
Another issue may be due to the code you're applying. Within a Foreach Loop Container, the first Script Task applies the following by passing to it a User::FileName variable:
public void Main()
if (File.Exists(Dts.Variables["FileName"].Value.ToString()) )
Dts.Variables["bolFileExists"].Value = true;
Dts.Variables["bolFileExists"].Value = false;
Leading from that are two line options. One applies the Expression "@bolFileExists==True" where the file is processed and the other applies "@bolFileExists==False" skipping the processing steps.