The expression green /white around the gills, meaning "somewhat pale, as from being sickly, nervous, or frightened", became popular from the early '30s but it is from the mid-1800. It probably refers to an earlier usage of gills meaning "face":
- When he heard how much the bill was, he looked a little green around the gills.
The notion of green to refer to looking ill is quite old: (Etymonline)
- From mid-13c. in reference to the skin or complexion of one sick.
The organ that enables most aquatic animals to take dissolved oxygen from the water. It consists of a series of membranes that have many small blood vessels. Oxygen passes into the bloodstream and carbon dioxide passes out of it as water flows across the membranes.
Green around the gills
- Sick-looking; pale and miserable; nauseated : He was looking green around the gills, so I told him to lie down
[1930s+; the date should probably be earlier; gills, ''face,'' is found by 1626]
pale around the gills
- Looking sickly or nauseous (1959+)
(The Dictionary of American Slang, Fourth Edition by Barbara Ann Kipfer, PhD. and Robert L. Chapman, Ph.D.)
The following extract from Wordwizard shows that similar expressions with "gills" referring to the lower part of the face were used in earlier centuries like "rosy about the gillS:
Green around/about the gills:
Looking ill, nauseated, pale and miserable, possibly from the effects of overeatng or motion sickness. “When she got off the rollercoaster she was feeling green around the gills.” A green complexion has signified illness since about 1300 and ‘gills’ carries the figurative meaning of the skin beneath the jaws and ears.
Rosy about the gills’ has meant being in good health since the late 17th century. Sir Francis Bacon used ‘red about the gills’ to signify anger (1626), whereas in the 19th century ‘white’ and ‘yellow’ about the gills meant looking ill. However, the alliterative green won out and surivives in the present day expression.
Blue around the gills is also sometimes used. And fishy about the gills refers to being hung over.
1902 “*She was shivering, and looked blue around the gills, as if half frozen.”–‘New York Times, 29 March, page SM6*.
1903 “. . . *and then he’d look green around the gills and sighed . . .”—‘Los Angeles Times,’ 3 July, page 16*
1932 “*The poor workman turned dark gree around the gills and began to tremble.”—‘Los Angeles Times,’ 10 July, page 15*
Note: In checking the quotes I came up with (not all of which are displayed above), it appears that "blue around the gills", which I was not familiar with, was used more than I had suspected, so that Spielberg’s only real transgression, if there is one, is in the use of the word IN which is never found, as far as I could determine, in combination with "gills".
(Cassell’s Dictionary of Slang, Facts on File Dictionary of Clichés, American Heritage Dictionary of Idioms, Chapman’s Dictionary of Slang, Picturesque Expressions by Urdang)