Thank you for bringing your question to the Community forum! It is likely that they will need to Whitelist Constant Contact. Here is an article that you can send to them, and they will be able to add Constant Contact as a trusted source so their security program won't try to block us!
If they receive this information, and you still are having trouble with bounces, particularly if they are showing up as "Suspended," please take a look at this article. If you email us at social_support(at)constantcontact(dot)com with your username, a reference to this post, and if you let us know who the contact is, (name, email address are helpful), we can look into the issue further as well.