While the restrictions on freedom of speech while in the employ of the government apply to contracted corporations and their employees, where the line between protected and unprotected speech is ambiguous, qualified immunity protects the government from claims of unlawful retaliation.
As the Fourth Amendment only protects against invasion by public actors, accessing of email records by employee of contracted corporation did not violate; the question of direction by state actors is ambiguous enough to warrant qualified immunity.
http://media.ca7.uscourts.gov/cgi-bin/rssExec.pl?Submit=Display&Path=Y2018/D06-20/C:17-2053:J:Easterbrook:aut:T:fnOp:N:2173897:S:0
Since 2011. An occasional commonplacing of the U.S. Circuit Courts of Appeal.
Neither an authoritative statement of the law nor legal advice. Quick work.