The simple answer to your question would be yes.
Super Saiyan Rosé as is said in the Dragon Ball Wikia, is the Super Saiyan Blue form of a true deity having his Super Saiyan 1 form exceed the power of a God.
What led me to believe this is this statement by Future Zamasu:
"So this is what happens when a Divine being surpasses Super Saiyan God, instead of turning blue they turn pink."
Also, considering the fact that a Kaioshin's apprentice is the successor to a Kaioshin, we can say that if Trunks would have become the next Kaioshin, he would have been able to access this form, given that he has achieved Super Saiyan God or has exceeded it's power with his Super Saiyan 1 form.
Talking about Goku and Vegeta, if either of them would have become a God of Destruction, having transformed into a real deity, they would have been able to access this form and the Blue counterpart would have ceased to exist.
Again, these statements of mine can be backed up from the wikia:
Future Zamasu gives a possible explanation of the form's color being due to a Saiyan possessing proper god status when they surpass Super Saiyan God, implying that it is likely that Goku, Vegeta, Vegito, or any mortal Saiyan Beyond God, could have obtained this form instead of Super Saiyan Blue had they held the position of a God such as Guardian (a position which Kami had once offered to Goku) or God of Destruction (as Whis considers them both to be candidates to replace Beerus, if he were to die) when they surpassed Super Saiyan God.
To discuss a bit on the comments posted on Gary Andrews30's answer,
Even though it was in a game, there are cases of non-Shinjin becoming Supreme Kais. An example being, Majin Ratopa being a former Supreme Kai in Dragon Ball Fusions.
This can be verified in the wikia here.
There would be a lot of difference in either of the aforementioned saiyans using Super Saiyan Blue or Super Saiyan Rosé. I'd like to back up this statement of mine using a few excerpts from the wikia.
This form, like its blue counterpart, provides Black with precise ki control, needed to properly use the form. Unlike its Super Saiyan Blue counterpart, the ki control of Super Saiyan Rosé is so precise and efficient, again due to Black's status as a true god, that this allowed him to create various weapons of energy at will. Unlike Super Saiyan Blue, which needed additional training to fully master it, this form does not display any sort of stamina decrease.