I dug a bit. Gamefaqs.com has a few entries for Crossfire and Cross Fire and got a lead on a Famicom title.
http://www.gamefaqs.com/console/nes/data/579848.htmlCross Fire KYG-C9 11/02/90 Kyugo (from famicomworld)
http://www.pinkgodzillagames.com/store/popup_image.php?pID=62&osCsid=d4from pinkgodzillagames:
Best described as a Contra clone, this side scrolling action game throws you into the middle of Vietnam with nothing more than your fists. Adding to the challenge, enemies are positioned in the background of the stages where they hurl grenades at your head. Watch out for the random helicopter fire or its "Gocchhhhaaa, Game Over."
HAL was a Japanese company and did port existing Famicom titles to the US. The notes at bottom seem to denote US version but that could mean in contrast to PAL, not as opposed to Famicom. Anyways this is my vote for what the heck it is

HAL has done arcade ports, but only of mega-hit games (Millipede, Joust, Stargate) so this rules out the only arcade game by the same name:
http://cgi.ebay.com/ORIGINAL-WILLIAMS-CROSSFIRE-ARCADE-GAME-PROMO-FLYER_W0QQitem......As if the game's nature itself wouldn't do just that

Not only that but the ROM may already have been dumped, I found a proto ROM here:
http://emu-russia.km.ru/default.php?a=s&sid=nes&query=Cross Fire
and a screenshot for you all.

So jajaja congrats it looks like a fairly good game from here! Certainly more respectable than shooting ball-bearings at an opponent ad infinitum