Would requiring criminal history on a job application for a government job or military enlistment be "violation of the Fifth Amendment?"
Of course not! That's because the 5th Amendment states, "nor shall be compelled in any criminal case to be a witness against himself." The 5th does not cover required disclosure outside of involuntary criminal (police/court) proceedings.
It is not a crime for a felon to try to obtain a weapon and truthfully answers the question on criminal history. It is only a crime if he/she lies (just as it would be for lying on government job applications).
There is already a Supreme Court case that says exactly that. Criminals are free not to disclose that they have a firearm.