Nick Ross took advantage of some hesitant defending to finish off Aaron Doran's cross midway through a one-sided first half but Inverness could not get the killer second and had a strong penalty appeal rejected when Momo Sissoko felled Billy McKay.
And Borja sickened the hosts when he shot low into the net following a low cross by James Dayton to earn his side the point that moved them into sixth in the Clydesdale Bank Premier League.
Shiels was back in the dugout following a four-match ban but he was sent off in injury-time for apparent dissent.
The Northern Irishman had used his rediscovered freedom to go into the away end before the game and shake the hands of the small group of visiting fans who had braved a snow-hit A9 to follow their team north.
There was only heavy rain and a swirling wind when they arrived but their team were soon hit by a blizzard of chances as Inverness set about ending a four-match losing streak, which included a William Hill Scottish Cup defeat at Rugby Park.
And the hosts missed the chance to move three points clear in second place when Borja netted the unlikely equaliser – before more drama as Shiels was sent off.