this is very strange.
If the Zip process is connected to the text file creation process by an 'on success' workflow it's not possible that the file creation fails and the ZIP process is executed.
An incorrect setup of the Sql server agent could cause issues when the DTS interacts with the file system but it seems that this is not the case because you tested different users.
You should do the following :
in enterprise manager open the sql server agent and go to Jobs
look for your job, right click, view job history
check 'show step details' , copy and check the messages from the job steps
If you have still problems getting a solution you could post :
the above messages
the DTS saving it as a Structured Storage File