Can I sue my employer for not giving me lunch breaks that are required according to our policy?

Violations of policies can be evidence of impermissible conduct, but are not usually the basis of a lawsuit all by themselves....  mainly, because employers reserve the right to alter policies, especially in Texas which is not thick with unions.  (If it was a union negotiated policy/contract-- the result could be a bit different)   Texas also does not require employers to give lunches or breaks... instead these breaks are usually part of a company's safety program to avoid any safety issues with OSHA. 

To answer your question, with no additional information other than the denial of a break, you could sue your employer, but you would not be successful in getting any relief from the lawsuit.  If you could show that the denial of breaks was for a different reason (like to harass the women and let the men go to lunch), then the policy violation could be evidence of discrimination-- and that's how policy violations are generally used in lawsuits.  If this is not applicable to your situation, then you could still fine a complaint with OSHA for the employer not giving breaks and creating an unsafe work environment.