A non-engaging abutment is used whenever there are multiple implant or abutment sites being restored with and being connected by the one prostetic assembly.
scenario
edetulous pt
implants at/near 2, 5, 8, 9, 12, 14
Dr wants implant bridge 2-8 and 9-14. Bridge to have the abutments and crowns as one piece, not separate abutments and the seat crowns on them. That is, he wants it screw retained instead of cemented/bonded.
You are joining three implant sites with each bridge. Non engaging is indicated so the path of insertion is not insanity inducing. The engaging part is the anti-rotational component that keeps the whole abutment from rotating which would loosen the screw. The other implants, being multiple sites on one restoration, keep a bridge, or even two crowns as one piece on two abutments(splinted) from rotating. If you are being required to make a bridge with one natural tooth and an implant as the abutments, a terrible idea but not always your call, and doc wants the implant screw retained the implant should probably be non-engaging for the same path of insertion issues as multiple implant sites being restored with screw retained prosthesis.
With engaging abutments the angle of all three implants would likely have to be less than .5 degrees from them all being perfectly parallel to get the bridge to seat.
That ain't gonna happen. As
@2thm8kr and
@grantoz were pointing out.
If you were doing separate abutments for the sites and then seating a bridge on the abutments as if they were normal preps, you would use engaging abutments since each abutment is not prevented from individually rotating by being joined one to another.
If that didn't come out clearly enough, feel free to ask specifically about what was unclear and I'll try differently.
This may help.
https://dentallabnetwork.com/forums/resources/implant-primer.12/