1) Nobody can be required to carry anything--just ain't constitutional, and gazillions of people would refuse, not to mention that any card is forgeable, so verification would always rest with some sort of computer database anyway.
2) A database would serve to distinguish citizens from non-citizens, and would consist ONLY of some reliable lifelong biometric record (possibly DNA) and an ID# for ease in locating/arranging the records. No name, no date of birth, absolutely no way for someone to use the database to identify individuals.
3) A phone line-based system, like the one now used for verifying credit cards, would enable anyone -- employers, airline security personnel, sellers of bulk ammonium nitrate, police, aviation schools, etc. -- to ascertain whether an individual's biometric is or is not in the database; i.e. to ascertain whether or not the individual is a citizen.
4) Absolute prohibition (with huge criminal penalties) on database searchers using a system which is capable of recording the biometric data (as that would enable matching of biometric data to other identifying information such as name, in transactions such as employment, or purchases by credit card or check, and thus enable the compiling of a identifying database of some portion of the citizenry).
5) Duplicate records would be held at the state level (state of birth or residence at naturalization, and at the individual's option state(s) of subsequent residence, and/or county of residence). This provides a check on federal meddling or incompetence in which individuals might either deliberately or accidentally be "de-citizenized", as it would be provided that any conflict between state and federal records is automatically ruled in favor of the individual.
This system would enable near-instantaneous confirmation of whether a person is or is not a citizen, without identifying the individual in question either to the questioner or to the maintainers of the database. This in turn would enable authorities and sensitive businesses to treat citizens and non-citizens differently, which it is currently impossible to do on a routine basis. If a person's citizenship can be determined quickly and non-intrusively, this would enable citizens to bypass closer scrutiny while applying higher standards of scrutiny to non-citizens. This wouldn't catch the Timothy McVeighs, but it would catch foreign terrorists, foreign drug runners, and illegal immigrants. It could also be helpful to citizens who look and/or sound "foreign", by providing a quick way to establish citizenship in situations where they may be suspected of being illegal immigrants, causing employers to fear hiring them, etc.
I see little point in even maintaining the concept of citizenship, if there's no practical way to determine who is or isn't one.
If you don't think that the gov does not maintain illegal databases, I'd like to sell you my house for a million bucks, t is worth it, trust me.