No legal issue with the break. Texas law does not require employers to offer breaks or lunches.
Are you so sure you passed the background check? Some employers do not perform the background check before hiring. They will first hire the candidate and then perform the background check. Sometimes it takes a few weeks to verify the criminal background before they can make an employment decision on it. They can't just run one of the online background checks and rely on that information. They must verify the felony conviction. If you are registered as a sex offender then it's highly unlikely your employer did not find that on a background check.
If you admitted to the conviction then I'm not sure you would even have a remedy against the employer even if the employer did not verify the conviction through a background check. If you are interested in pursuing this issue, contact a local employment lawyer.