Do construction employers have the legal right to monitor my computer activity?

Yes, Federal law here in the U.S. clearly states that the computer system is the property of the construction employer and as such the construction employer has the right to monitor all email activity and Internet activity.

If your email is being monitored and if you send a message that violates your construction employer's policy and you end up terminated for that message, you'd have a real hard time making a claim of invasion of privacy stick.