Can a Soldier be flagged for a civilian investigation?

Yes, it is a law enforcement flag (technical term is Suspension of Favorable Action-SFPA). It is an administrative tool, not a punishment. A flag of this nature is non-transferrable so it communicates to HRC not to move the Soldier until the issue is resolved. Once removed, there is no future indicator on the Soldier's record that he ever had it, so again, it is not punitive in nature. If the Soldier becomes eligible for promotion (advancement or meets cut-off) then the promotion would be held in abeyance until the issue is resolved. If the flag is removed in the Soldier's favor, the promotion can proceed. See your BN/BDE S1 for more details (and AR 600-8-2 as indicated by SGM Quick who always beats me to the draw!)

Have you researched AR 600-8-2, Suspension of Favorable Personnel Actions (Flags)?

http://www.apd.army.mil/pdffiles/r600_8_2.pdf

That is part of clearing an installation.