"I am an exempt employee. Do I have to take a lunch break?"
*** If required by company policy (likely, see below), yes.
CT laws say "Under 31-51ii of the State Statutes, no person shall be required to work for seven and one-half or more consecutive hours without a period of at least thirty consecutive minutes for a meal. The Labor Commissioner shall exempt any employer from the requirements if (1) requiring compliance would be adverse to public safety, (2) the duties of the position may only be performed by one employee, (3) the employer employs less than 5 employees on a shift at a single place of business (only applies to that shift), (4) the continuous nature of an employer's operation requires employees be available to respond to urgent conditions and employees are compensated for break and meal periods."
I would imagine that a CT based company would require that you take the break to prevent any confusion as to whether they allowed it or not.
"I am required to work 8 hours. I stay at my desk during lunch and continue to work until my eight hours is completed."
*** And presumably, your issue is that they expect (and demand) your 8 working hours PLUS the expected lunch break each day.
"If I am exempt why should they care if I take a lunch?"
*** See the above state law.
"It seems an exempt employee mean all the benefits are in the employers corner and none for the employee!"
*** If you don't like the policy as established by your employer, you are certainly allowed to find another employer who may be more amenable to your expectation of a 'working lunch'.