You'll want to refer to Nebraska statute 28-1204:
(1) Any person under the age of eighteen years who possesses a handgun commits the offense of unlawful possession of a handgun.
(2) This section does not apply to the issuance of handguns to members of the armed forces of the United States, active or reserve, National Guard of this state, or Reserve Officers Training Corps, when on duty or training, or to the temporary loan of handguns for instruction under the immediate supervision of a parent or guardian or adult instructor.
(3) Unlawful possession of a handgun is a Class I misdemeanor
and 28-1204.01
(1) Any person who knowingly and intentionally does or attempts to sell, provide, loan, deliver, or in any other way transfer the possession of a firearm to a juvenile commits the offense of unlawful transfer of a firearm to a juvenile. The county attorney shall have a copy of the petition served upon the owner of the firearm, if known, in person or by registered or certified mail at his or her last-known address.
(2) This section does not apply to the transfer of a firearm, other than a handgun, to a juvenile:
(a) From a person related to such juvenile within the second degree of consanguinity or affinity if the transfer of physical possession of such firearm does not occur until such time as express permission has been obtained from the juvenile's parent or guardian;
(b) For a legitimate and lawful sporting purpose; or
(c) Who is under direct adult supervision in an appropriate educational program.
(3) This section applies to the transfer of a handgun except as specifically provided in subsection (2) of section 28-1204.
(4) Unlawful transfer of a firearm to a juvenile is a Class III felony.
So... 18-20 year olds can legally possess a handgun as long as it is not concealed. They can legally get the handgun from the relationships mentioned in one of the earlier posts.
As long as they are 18+, the same transport rules would apply as they do to you. Without a CHP, transport would need to be open in a vehicle, or unloaded and separated from the ammunition (in the trunk).