They are right.. it doesn't have to do with the server but your site(s).
Odds are you have an outdated or an unsecured script in place.

Most of the time it is some type of formail but it could by any script.. php or anything else. You have to go through your logs and see what has been hacked and plug the hole. Always make sure all scripts and forms up kept updated.

Your hosting might be able to help pin it down for you if you are not used to that kind of work. Or you can hire a developer to take a peek and fix it for you.

Then.. when done... your google error should be removed.
Also.. the google security warning should include some information what the issue is.