Is it okay to have a chest x-ray without covering your genitals?

Probably: If the machine is up to date, it will have a well focused beam of lower intensity due to electronic recording rather than xray film. The genitals are not in the beam - it is possible that there might be a little scatter but this would be extremely little. The only real concern, even with older machines, is alterations in the genetics of sperm , if you are no longer reproducing, that isnt a worry.

Yes, but: Best practice would be to shield your genitals anytime you receive an x-ray or ct scan. The radiation dose is small, but accumulative over your lifetime. You should ask the technologist performing the study for a shield.

Keep covered: Amount of radiation decreases with distance organ is away from xray beam. However any radiation exposure that is not needed is unwise. Keep lead over gonads when not body part necessary to be visualized for diagnosis.

It never hurts: to maximize radiation safety, but practically speaking the pelvis receives virtually no scatter radiation from a properly collimated chest xray.