First a clarification. If one insists that "knowing" requires self awareness and intelligence, then the Juno spacecraft of course doesn't "know" anything. Rather than getting hung up on the silliness of what "knowing" means, it's better to look for an alternative way of answering the question. That alternative: How sophisticated is Juno's onboard computer software? Did the spacecraft software know, in some limited way, that it had indeed achieved orbit?
I'm making this community wiki because this is not an answer. The correct answer is very hard to find. (As far as I can tell, none of the proferred answers are correct.) Both JPL and Lockheed Martin release very little, if any, technical details on their vehicles' inner workings. The vehicle's guidance, navigation, and control systems and mission managers are apparently stamped as ITAR restricted and as proprietary. I'll look at two cases, one in which the vehicle is not aware that it is in orbit about Jupiter and the other in which it is.
One way that the vehicle could have performed the orbit insertion burn autonomously would have been to orient the vehicle in a predetermined orientation, start the rocket, and stop when delta V to go (the difference between a predetermined desired delta V and accumulated sensed delta V) reached zero. Suppose both the predetermined orientation and predetermined desired delta V had been a part of a command sequence sent to the spacecraft from Earth. If this is the case, the vehicle did not know it was in orbit. All it knew was that the burn was complete. The software to do this is very simple. If simple software is good enough to do the job, simple is best.
A good deal more sophistication might be needed if that simple approach would not suffice. For example, the spacecraft might need more sophisticated guidance, navigation, and control software, and more complex mission management software. (It apparently has rather extensive failure detection, isolation, and recovery software). In this more sophisticated version, the mission manager may well have a on-orbit mode, the transition to which is triggered by the GNC software assessing that the desired orbit has been achieved. While this is not self-aware software, it is software that is aware that the vehicle is indeed in orbit about Jupiter.
Note that Juno is controlled by a 200 megahertz RAD750 computer with 128 megabytes of memory and 256 megabytes of flash storage. That's the equivalent of a low cost personal computer from 1999. There's not much room in that rather limited system for very much sophistication.