Just as military camouflage can be legally worn by civilians on most jurisdictions (e.g., in the U.S.), can I create a stealth aircraft for non-military use? More precisely, can I register and fly the thing?
Yes you may, but you need to carry a mode C transponder while above 10,000 ft or in some types of controlled airspace and have it switched on. Which pretty much negates the stealth capabilities of your aircraft.
To quote the linked AOPA page:
According to 14 CFR 99.13, no person may operate an aircraft into or out of the United States, or into, within, or across an ADIZ designated in subpart B unless operating a transponder with Mode C. Certain exemptions might apply to aircraft that were not originally certified with an engine-driven electrical system; see 99.13(d).