I have been trying to interpret it myself and find the wording too absolute. It simply says it may reroll 1 die. So no matter what source the squadron recieves the reroll from, it may reroll 1 die. No more.
Now I agree that that would be a too strickt interpretation, I do agree that you could reroll a dice for each BCC in range, in combination of other abilities and rerolling rerolls.
It is worth 8 points, so you may expect a lot from it.
Sure you get 1 reroll per card, but is it per card since they are different upgrades due to being on a different ship? Or is it just the 1 time.
I believe in the first definition. Since one can aggressively reroll dice from multiple sources, why can't the a card with the service name on a different ship work? They are after all a While effect and the rule for While effects is worded to work 1 at a time due to the "A 'While' effect..." wording.