This M14 History is exhaustive.
http://www.milsurps.com/showthread.php?t=4323
Shughart's M14 gets a mention on page 74. All it says is "M14 w/ leather sling and dot sight".
There's a good chance that Shughart used a Leupold MK 4 / Ultra since that reached Special Forces units 1988 for the newer M24 system and was in use Army wide by 1992.
Edit: the movie is littered with understandable anachronisms and inaccuracies such as M4's, the near total absence of 6 color desert uniforms and of course Aimpoints.
M1 and M3 scopes are made by various airsoft manufacturers at a variety of price points.