Nathan Young wrote:

If so, what action should I have taken?


Take them up on their suggestion to call the next radar facility. You
do not suddenly lose your transition rights because you got terminated.



I have always assumed that VFR flight following does not provide
transition services to facilities that aren't on their field.

You're wrong, it does.


Hence,
I would have called McEntire's Class D.

A courtesy call wouldn't hurt.



I run into this getting flight following around the OHare Bravo, and
crossing through DuPage's Class D. I always suggest to the approach
controller that I get a freq change to call DuPage tower, and then
give him/her a call on the other side.

Total waste of time.