If the carrier is telling you that the load must go on, no matter if you're sick or not, point them towards this little DOT regulation:
§392.3 Ill or fatigued operator.
No driver shall operate a motor vehicle, and a commercial motor carrier shall not require or permit a driver to operate a commercial motor vehicle, while the driver's ability or alertness is so impaired, or so likely to become impaired, through fatigue, illness, or any other cause, as to make it unsafe for him/her to begin or continue to operate the commercial motor vehicle. However, in a case of grave emergency where the hazard to occupants of the commercial motor vehicle or other users of the highway would be increased by compliance with this section, the driver may continue to operate the commercial motor vehicle to the nearest place at which that hazard is removed.
[35 FR 7800, May 21, 1970, as amended at 60 FR 38746, July 28, 1995].
http://www.fmcsa.dot.gov/rules-regul...ction_toc=1803
The interpretation:
§392.3 Ill or Fatigued Operator
Question 1: What protection is afforded a driver for refusing to violate the FMCSRs?
Guidance: Section 405 of the STAA—Surface Transportation Assistance Act of 1982 (49 U.S.C. 31105) states, in part, that no person shall discharge, discipline, or in any manner discriminate against an employee with respect to the employee's compensation, terms, conditions, or privileges of employment for refusing to operate a vehicle when such operation constitutes a violation of any Federal rule, regulation, standard, or order applicable to CMV safety. In such a case, a driver may submit a signed complaint to the Occupational Safety and Health Administration.